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1. Available online at www.sciencedirect.com
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Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
www.elsevier.com/locate/cma
Isogeometric Analysis for second order Partial Differential
Equations on surfaces
Luca Dedèa,∗, Alfio Quarteronia,b
a CMCS—Chair of Modeling and Scientific Computing, MATHICSE, École Polytechnique Fédérale de Lausanne, Station 8, Lausanne, CH-1015,
Switzerland
b MOX—Modeling and Scientific Computing, Mathematics Department “F. Brioschi”, Politecnico di Milano, via Bonardi 9, Milano, 20133, Italy1
Available online 18 November 2014
Abstract
We consider the numerical solution of second order Partial Differential Equations (PDEs) on lower dimensional manifolds,
specifically on surfaces in three dimensional spaces. For the spatial approximation, we consider Isogeometric Analysis which fa-
cilitates the encapsulation of the exact geometrical description of the manifold in the analysis when this is represented by B-splines
or NURBS. Our analysis addresses linear, nonlinear, time dependent, and eigenvalues problems involving the Laplace–Beltrami
operator on surfaces. Moreover, we propose a priori error estimates under h-refinement in the general case of second order PDEs on
the lower dimensional manifolds. We highlight the accuracy and efficiency of Isogeometric Analysis with respect to the exactness
of the geometrical representations of the surfaces.
c
⃝ 2014 Elsevier B.V. All rights reserved.
Keywords: Second order Partial Differential Equations; Manifolds; Surfaces; Laplace–Beltrami operator; Isogeometric Analysis; A priori error
estimation
1. Introduction
In several instances, Partial Differential Equations (PDEs) are set up on lower dimensional manifolds with respect
to the hosting physical space, namely on surfaces in three dimensions or curves in two or three-dimensions [1]. Ap-
plications include problems in Fluid Dynamics, Biology, Electromagnetism, and image processing as reported for
example in [2–6]. In addition, PDEs on lower dimensional manifolds could be obtained as reduced mathematical
formulations of PDEs defined in thin geometries, e.g. for plates and shells structures [7].
The numerical approximation of these PDEs generally requires the generation of an approximated geometry com-
patible with the analysis, as it is the case for the Finite Element method (see e.g. [8–10]). In particular, the approxi-
mation of the curvature of surfaces may significantly affect the total error associated to the numerical approximation.
∗ Corresponding author. Tel.: +41 21 6930318; fax: +41 21 6935510.
E-mail address: luca.dede@epfl.ch (L. Dedè).
1 On leave.
http://dx.doi.org/10.1016/j.cma.2014.11.008
0045-7825/ c
⃝ 2014 Elsevier B.V. All rights reserved.

2. 808 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Typically, schemes based on the Finite Element method have been used for the approximation of PDEs on surfaces
with particular emphasis in controlling and limiting the propagation of the errors associated to the discrete geometri-
cal representation. With this aim, surface Finite Element methods [11,12] and geometrically consistent Finite Element
mesh adaptations [13,14] have been considered. As alternatives, approaches based on the implicit or immersed sur-
faces have been proposed, namely based on level set formulations [4,15] or diffuse interfaces strategies [5]. Still, for
a broad range of geometries (surfaces) of practical interest, the above mentioned approaches are not error free in the
geometrical representation.
As alternative to these approaches, in this paper we propose numerical approximation of PDEs on lower dimen-
sional manifolds by means of Isogeometric Analysis. Our approach is motivated by the fact that a broad range of
geometries of practical interest are exactly represented by B-splines or NURBS [16].
Isogeometric Analysis is an approximation method for PDEs based on the isoparametric concept for which the
same basis functions used for the geometrical representation are then also used for the numerical approximations of the
PDEs [17,18]. Typically, B-splines or NURBS geometrical representations are considered for Isogeometric Analysis,
even if, more recently, T-splines [19] have been successfully utilized. Since NURBS are the golden standard in Com-
puter Aided Design (CAD) technology, the use of Isogeometric Analysis facilitates the encapsulation of the exact geo-
metrical representation in the analysis and simplifies the establishment of direct communications between design and
numerical approximation of the PDEs. Moreover, NURBS-based Isogeometric Analysis possesses several advantages
besides the geometrical considerations, especially in terms of smoothness of the basis functions and accuracy proper-
ties [20–22]. Nowadays, Isogeometric Analysis have been successfully used in a broad range of applications in com-
putational mechanics and optimization, see e.g. [17,23–26]. In particular, Isogeometric Analysis have been considered
for solving shell problems, as e.g. in [27], and, more recently, Isogeometric Analysis in the framework of the Boundary
Element method [28] has been used to take advantage of the exact geometrical representation of surfaces [29].
In this work we provide for the first time a general formulation of the numerical approximation of second order
PDEs defined on lower dimensional manifolds described by NURBS, specifically surfaces, by means of Isogeometric
Analysis. We discuss the representation of the manifolds in a general framework by means of geometrical mappings
from the parameter space to the physical domain; consequently, in view of the use of Isogeometric Analysis based
on the Galerkin method [10], we recast the weak forms of the problems and the spatial differential operators in the
parameter space. We provide a priori error estimates under h-refinement for the numerical approximation by means of
Isogeometric Analysis, thus extending the results of [20,21] to the case of the second order PDEs on lower dimensional
manifolds; with this aim, an interpolation error estimate for the NURBS space on the manifold is proposed. We show
the accuracy and efficiency of the method by solving several PDEs endowed with the Laplace–Beltrami spatial opera-
tor on surfaces. In particular, as few remarkable instances, we address the numerical solution of the Laplace–Beltrami
problem, the eigenvalue problem, a time dependent linear advection–diffusion equation, and the Cahn–Allen phase
transition equation [30,31]. For both the Laplace–Beltrami and the eigenvalue problems we compare the convergence
rates of the errors obtained by means of Isogeometric Analysis with those expected from the a priori error estimates
and we highlight the advantages of exactly representing the geometries at the coarsest level of discretization. In this
respect, we also compare the numerical results obtained by means of Isogeometric Analysis with those obtained by
the standard isoparametric Finite Element method [9].
This work is organized as follows. In Section 2 we discuss the representation of lower dimensional manifolds by
NURBS and the role of the parametrization in the definition of geometrical mappings. In Section 3 we consider the
PDEs on the manifolds for problems involving the second order Laplace–Beltrami spatial operator. In Section 4 we
discuss the numerical approximation schemes, specifically Isogeometric Analysis for the spatial approximation; for
the time dependent problems, the generalized-α method [32] is considered and a SUPG stabilization scheme [33] is
presented because of its suitability to treat advection dominated problems. In Section 5 we provide the interpolation
error and a priori error estimates for h-refined NURBS “meshes”. In Section 6 we report and discuss the numerical
results for PDEs on surfaces. Final considerations are reported in the Conclusions.
2. Manifolds represented by NURBS
In this section we introduce in an abstract setting lower dimensional manifolds in the physical space, e.g. curves
and surfaces, represented by suitable geometrical mappings. We recall the definition of generic functions and their
derivatives on the manifold and we express them in terms of the parametric coordinates upon which the geometrical

3. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 809
mapping is built. Finally, we specifically select manifolds defined by B-splines and NURBS and we briefly recall the
basics of these geometrical representations.
2.1. Manifolds and geometrical mapping
Let us consider a generic (Riemannian) manifold, say Ω ⊂ Rd with d ≥ 1, defined in the physical space Rd [1].
Let us assume that the manifold is obtained by means of a geometrical mapping from a parameter space, say Rκ,
into the physical space Rd, with d ≥ κ ≥ 1. If d > κ the manifold is lower dimensional with respect to the
physical space. More precisely, given a parameter domain, say 
Ω ⊂ Rκ, and a vector-valued independent variable
ξ = (ξ1, . . . , ξκ) ∈ Rκ, the manifold Ω ⊂ Rd is defined by means of the geometrical mapping:
x : 
Ω → Rd
, ξ → x(ξ). (1)
For example, if d = 3 and κ = 1, the manifold Ω represents a curve in R3, while if κ = 2, Ω is a surface in R3.
Specifically, we consider compact, connected, and oriented manifolds Ω defined from parameter domains 
Ω of finite,
positive measure with respect to the topology of Rκ (0 < |
Ω| < +∞).
For the mapping (1), we define its Jacobian:

F : 
Ω → Rd×κ
, ξ → 
F(ξ), 
Fi,α(ξ) :=
∂xi
∂ξα
(ξ) i = 1, . . . , d, α = 1, . . . , κ. (2)
High order differentials of the geometrical mapping (1) are multidimensional arrays 
F(m) (tensors of order m + 1)
defined, for m ≥ 1, as:

F(m)
: 
Ω → Rd×κ×···×κ
, ξ → 
F(m)
(ξ),

F
(m)
i,α1,...,αm
(ξ) :=
∂mxi
∂ξα1 · · · ∂ξαm
(ξ) i = 1, . . . , d, α1, . . . , αm = 1, . . . , κ;
(3)
we observe that the Jacobian 
F corresponds to m = 1, i.e. 
F ≡ 
F(1), while the Hessian of the mapping is obtained for
m = 2. We also introduce the first fundamental form of the mapping from the Jacobian 
F (2):

G : 
Ω → Rκ×κ
, ξ → 
G(ξ), 
G(ξ) :=


F(ξ)
T 
F(ξ) (4)
and finally its determinant:

g : 
Ω → R, ξ → 
g(ξ), 
g(ξ) :=

det


G(ξ)

. (5)
We observe that in the case for which κ = d, then 
F(ξ) ∈ Rd×d and 
g(ξ) ≡ det


F(ξ)

(when positive). We assume
that the geometrical mapping (1) is “sufficiently” smooth, e.g. C1(
Ω), and invertible a.e. in 
Ω. Notice that we allow
the mapping to be locally not invertible in subdomains Q of Ω with zero measure in the topology of Rκ; specifically,
we require that 
g(ξ) > 0 a.e. in 
Ω with 
g(ξ) = 0 for ξ ∈ 
Q ⊂ 
Ω only if meas


Q

≡ 0, where 
Q is the subdomain
of 
Ω mapping the subdomain Q of Ω.
Finally, in view of the derivation of the a priori error estimate in Section 5, we introduce, following Eqs. (2), (4)
and (5), the Jacobian, the differential of order m, the first fundamental form of the mapping, and its determinant in the
manifold Ω as, respectively2:
F : Ω → Rd×κ
, x → F(x), F(x) := 
F(ξ) ◦ x−1
(ξ), (6)
F(m)
: Ω → Rd×κ×···×κ
, x → F(m)
(x), F(m)
(x) := 
F(m)
(ξ) ◦ x−1
(ξ), (7)
G : Ω → Rκ×κ
, x → G(x), G(x) := 
G(ξ) ◦ x−1
(ξ), (8)
g : Ω → R, x → g(x), g(x) := 
g(ξ) ◦ x−1
(ξ). (9)
where we used the geometrical mapping (1).
2 In this paper we opted to indicate the geometrical mapping and its gradient as x and F, respectively, in order to use a notation typical of
PDEs defined on lower dimensional manifolds. Conversely, we remark that in the standard notation for Isogeometric Analysis, the geometrical
mapping (1) is indicated as F and the Jacobian as ∇F (see [20]).

4. 810 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
2.2. Functions and differential operators on manifolds
Let assume that a “sufficiently” regular function is defined on the manifold Ω, e.g. φ ∈ C0(Ω) for all x ∈ Ω; then,
since we consider invertible geometrical mappings, we can write:
φ(x) = 
φ(ξ) ◦ x−1
(ξ), (10)
where 
φ(ξ) := φ (x(ξ)). Moreover, we define the gradient on the manifold Ω of the function φ ∈ C1(Ω), say
∇Ω φ ∈ Rd, as the projection of the gradient operator associated to the physical space onto the manifold. With this
aim, we introduce the smooth prolongation of the function φ(x) from 
Ω into a tubular region in Rd containing 
Ω
[13,34], say 
φ(x), and the projector tensor P(x) ∈ Rd×d for which we have:
∇Ω φ(x) :=

P(x) ∇
φ(x)

for x ∈ Ω. (11)
For example, for a curve in Rd we have ∇Ω φ(x) =

∇
φ(x) · tΩ (x)

tΩ (x), where tΩ (x) is the unit tangent vector to
the curve, being P(x) = tΩ (x) ⊗ tΩ (x) for x ∈ Ω. For a surface in Rd (d = κ + 1), the projector tensor reads:
P(x) := I − nΩ (x) ⊗ nΩ (x) for x ∈ Ω, (12)
with the unit vector nΩ (x) normal to the manifold Ω in Rd and I the identity tensor in Rd×d; in this case, we obtain
that ∇Ω φ(x) = ∇
φ(x) −

∇
φ(x) · nΩ (x)

nΩ (x).3 Finally, we introduce the Laplace–Beltrami operator associated to
the manifold Ω for a function φ ∈ C2(Ω) as:
∆Ω φ(x) := ∇Ω · (∇Ω φ(x)) , (13)
where the divergence operator ∇Ω · is defined as ∇Ω · v(x) = trace [∇Ω v(x)] for all v ∈

C1(Ω)
d
. By using the
notation of Eq. (11), we obtain for the Laplace–Beltrami operator (see [1]):
∆Ω φ(x) = trace

P(x) ∇2
φ(x) P(x)

for x ∈ Ω, (14)
where ∇2· indicates the Hessian operator such that

∇2
φ(x)

i, j
:= ∂2
φ
∂xi ∂xj
(x) for i, j = 1, . . . , d.
By using the geometrical mapping (1) and following [35], the gradient on the manifold (11) can be rewritten as:
∇Ω φ(x) =


F(ξ) 
G−1
(ξ)
∇
φ(ξ)

◦ x−1
(ξ), (15)
where 
∇
φ : 
Ω → Rκ is the gradient operator in the parameter space. Similarly, for the Laplace–Beltrami operator of
Eq. (13), we have [35]:
∆Ω φ(x) =

1

g(ξ)

∇ ·


g(ξ) 
G−1
(ξ) 
∇
φ(ξ)

◦ x−1
(ξ). (16)
High order derivatives of the function φ on the manifold Ω are indicated as ∇m
Ω φ with m = 1, 2, . . . , with the gradient
on the manifold ∇Ω φ corresponding to m = 1; we remark that, in a similar manner to [36], ∇m
Ω φ indicates the vector
of all the mth order derivatives of φ on Ω. Finally, in view of the definitions of integrals in the weak form of the PDEs,
the differential dx (or dΩ) can be written as dx = 
g(ξ) dξ (or dΩ = 
g(ξ) d
Ω).
2.3. Geometrical mapping by NURBS
We assume that the geometrical mapping introduced in Eq. (1) defines a manifold Ω represented by either B-splines
or NURBS; for a detailed description we refer the reader to [16]. We observe that in the framework of Isogeometric
Analysis, the choice of T-splines [19] represents a valid alternative and generalization.
3 Notice that for a surface in R3 the unit normal vector nΩ (x) is obtained by mapping
nΩ (ξ) :=

tΩ,1(ξ)×

tΩ,2(ξ)
∥

tΩ,1(ξ)×

tΩ,2(ξ)∥
with
tΩ,α(ξ) := ∂x
∂ξα
(ξ) for
α = 1, 2.

5. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 811
Fig. 1. Univariate, globally C p−1-continuous, B-splines basis functions


Ni (ξ)
n
i=1 of degrees p = 1, 2, and 3 obtained by the knot vectors
Ξ =

{0}p+1 , 1
5 , 2
5 , 3
5 , 4
5 , {1}p+1

, respectively; 
Ω = (0, 1).
The geometrical mapping (1) represented in terms of NURBS reads:
x(ξ) =
nbf

i=1

Ri (ξ) Pi , (17)
where 
Ri (ξ) are the NURBS basis functions defined in the parameter domain 
Ω and Pi ∈ Rd are the control points in
the physical space for i = 1, . . . , nbf . The NURBS basis functions 
Ri (ξ) are defined from B-splines basis functions

Ni (ξ) and weights wi ∈ R as:

Ri (ξ) :=
wi
nbf

i′=1
wi′ 
Ni′ (ξ)

Ni (ξ) for i = 1, . . . , nbf ; (18)
we observe that B-splines geometries in Rd can be seen as a particular case of NURBS with weights equal to the
unity.4 The (multivariate) B-splines basis functions 
Ni (ξ) are obtained by tensor product rule of univariate B-splines
basis functions, say 
Nα,i (ξα) for i = 1, . . . , nbf,α, where α = 1, . . . , κ indicates the parametric direction ξα in Rκ
with ξ = (ξ1, . . . , ξκ) and nbf = Π κ
α=1nbf,α. The univariate B-splines basis functions 
Nα,i (ξα) are recursively built
by using the Cox–de Boor recursion formula starting from a knot vector Ξα :=

ξα, j
nbf,α+pα+1
j=1 with ξα, j ∈ R which,
together with the polynomial degree pα, completely characterizes the properties of the basis functions. We indicate
the minimum polynomial degree of the basis functions as p := minα=1,...,κ pα; in several instances, pα = p for all
α = 1, . . . , κ.
For example, in Fig. 1 we report open-knot, univariate B-splines basis functions of degree p = 1, 2, 3 obtained
by the knot vectors Ξ =

{0}p+1
, 1
5 , 2
5 , 3
5 , 4
5 , {1}p+1

, respectively; we remark that the basis functions are globally
C p−1-continuous in 
Ω = (0, 1) for this specific choice of the knot vector.
We observe that the parameter domain 
Ω is obtained by the knot vectors Ξα for α = 1, . . . , κ as 
Ω =
(ξ1,1, ξnbf,1+p1+1) × · · · × (ξκ,1, ξnbf,κ +pκ +1), which defines a NURBS patch; by convention, we consider the case
for which 
Ω = (0, 1)κ, i.e. ξα,1 = 0 and ξnbf,α+pα+1 = 1 for all α = 1, . . . , κ. The tensor product of the knots
also defines a partition of the parameter domain 
Ω into subdomains 
Ωe, for e = 1, . . . , nel, which we regard as
“mesh” elements in the parameter domain, nel being their number; the set of these elements, i.e. the “mesh” of the
parameter domain, is denoted as 
Th. The geometrical mapping (17) of the elements 
Ωe into the physical space defines
the elements Ωe for e = 1, . . . , nel and, similarly, the corresponding “mesh” in the physical domain Th. In addition,
we introduce the notation 

Ωe to denote the support extension of 
Ωe, i.e. the union of the supports of the basis functions

Ri (ξ) for i = 1, . . . , nbf whose supports possesses an intersection with 
Ωe of nonzero measure in the topology of
Rκ [20]; the mapping of the support extension in the parameter domain 

Ωe into the physical space is denoted by 
Ωe.
4 NURBS are obtained by projective transformations of B-splines defined in the physical space Rd+1.

6. 812 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Fig. 2. Illustration of a geometrical mapping x(ξ) from the parameter domain 
Ω into the manifold Ω (physical domain), a mesh element 
Ωe and
its support extension 

Ωe in the parameter domain, and the corresponding mesh element Ωe and support extension 
Ωe into the physical space; the
example refers to a B-splines or NURBS surface obtained with a basis of degree p = 2 without internally repeated knots yielding nel = 49 mesh
elements.
Fig. 3. NURBS surfaces Ω in R3: cylindrical shell of radius 1 and height 4 (left) and spherical shell of radius 1 (right). The mesh elements are
highlighted in black, the control net and control points {Pi }
nbf
i=1, in red. (For interpretation of the references to color in this figure legend, the reader
is referred to the web version of this article.)
We schematically depict these concepts in Fig. 2. Finally, we indicate with 
he the characteristic size (diameter) of the
element 
Ωe and with 
h the global “mesh” size 
h := max


he : 
Ωe ∈ 
Th

; correspondingly, we define the characteristic
size (diameter) of the element Ωe in the physical domain from Section 2.2 as:
he := ∥
F∥L∞(
Ωe)

he ∀e = 1, . . . , nel (19)
and the global “mesh” size h := max {he : Ωe ∈ Th}. We observe that the definition of he coincides with the one given
in [20] for κ = d.
In Fig. 3 we report two examples of surfaces in R3 obtained by NURBS, a cylindrical and a spherical shell (we
refer the reader to e.g. [16,17] for the construction of these geometries). We observe that both the geometries are
exactly represented by NURBS at the coarsest level of discretization nel = 4 and 8 mesh elements and nbf = 18
and 45 basis functions for the cylinder and the sphere with p = 2, respectively. We remark that the standard
geometrical representation of the sphere with bivariate NURBS basis functions is built starting from the knot vectors
Ξ1 =

{0}3, 1
4 , 1
2 , 3
4 , {1}3

and Ξ2 =

{0}3, 1
2 , {1}3

; accordingly, the geometrical mapping x is singular at the two

7. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 813
poles of the sphere. In several circumstances, more complicated geometries can be generated by combining multiple
NURBS patches.
3. PDEs on the manifold
With the aim of generality, we introduce a nonlinear parabolic PDE with second order spatial derivatives
corresponding to the Laplace–Beltrami operator. Then, as particular cases of such general PDE, we present the
equations we will address in this work, namely: an elliptic PDE, an eigenvalue problem, a time dependent
advection–diffusion equation and the Cahn–Allen equation defined on the manifold. Then, in view of the numerical
approximation, we rewrite the weak form of the equations in the parametric space by means of the geometrical
mapping introduced in Section 2.
3.1. A nonlinear parabolic PDE with Laplace–Beltrami operator
Let us denote with Γ = ∂Ω the boundary of the manifold Ω. We say that Ω possesses a boundary if meas (Γ) ̸= 0
in the topology of Rκ−1; for example, by referring to Fig. 3(left), the cylindrical shell has a boundary Γ =

x = (x, y, z) ∈ R3 : x2 + y2 = 1 and z = {0, 4}

, while the sphere does not possess a boundary (meas (Γ) = 0). If
meas (Γ) ̸= 0, then we can partition it into two non overlapping subdomains, say ΓD and ΓN , such that ΓD ∪ ΓN ≡ Γ
and
◦
Γ D ∩
◦
Γ N = ∅.
For the sake of simplicity and unless when necessary for clarity, we will omit from here on the explicit dependence
of the functions defined in the physical space Rd on the spatial variable x and, similarly, for the functions in the
parameter space Rκ, the explicit dependence on the independent variable ξ.
By recalling the definition of the Laplace–Beltrami operator (13), we assume the following expression for the time
dependent nonlinear parabolic PDE, where u(t) represents its solution:
ν
∂u
∂t
(t) − µ∆Ω u(t) + V · ∇Ω u(t) + σ (u(t)) = f in Ω × (0, T ),
u(t) = γ on ΓD × (0, T ),
µ∇Ω u(t) · nΓ = 0 on ΓN × (0, T ),
u(0) = uin in Ω × {0},
(20)
where T > 0 is the final time and nΓ is the unit vector normal to the boundary Γ (note that nΓ ·nΩ = 0 for all x ∈ Γ).
For simplicity, the coefficients ν ≥ 0 and µ > 0 are assumed as constants in R, the advection field V ∈

L∞(Ω)
d
is such that V · nΩ = 0, ∇Ω · V ∈ L2(Ω) and ∇Ω · V = 0 a.e. for x ∈ Ω, and the source term f ∈ L2 (Ω); the
reaction term σ is assumed “sufficiently” regular, i.e. as σ ∈ L∞ (Ω × (0, T )). Moreover, γ ∈ H1/2(ΓD), and the
initial condition uin ∈ L2(Ω); for the sake of simplicity, we assumed a homogeneous Neumann condition on ΓN . We
observe that, if the manifold Ω does not possesses a boundary (meas (Γ) = 0), the boundary conditions in Eq. (20)
become meaningless and they should be replaced by other conditions on the solution u(t) depending on the choice of
the data (see Section 3.2).
In view of the weak form of the problem (20), we introduce the affine manifold Vγ and the function space V as:
Vγ :=

v ∈ H1
(Ω) : v|ΓD
= γ

, V :=

v ∈ H1
(Ω) : v|ΓD
= 0

. (21)
Moreover, we introduce the following forms and functionals:
a(v, w) :=

Ω
µ ∇Ω v · ∇Ω w dΩ, b(v, w) :=

Ω
v V · ∇Ω w dΩ,
c(w)(v) :=

Ω
v σ(w) dΩ, m(v, w) :=

Ω
v w dΩ,
q(v) :=

Ω
v f dΩ,
(22)
and the weak residual:
Res (w(t)) (v) := q(v) − a(v(t), w(t)) − b(v, w(t)) − c(w(t))(v) − ν m

v,
∂w
∂t
(t)

, (23)

8. 814 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
for any v ∈ V, w(t) ∈ Vγ , for almost all t ∈ (0, T ). Then, the weak form of problem (20) reads, for almost all
t ∈ (0, T ):
find u(t) ∈ Vγ : Res (u(t)) (v) = 0 ∀v ∈ V,
with u(0) = uin.
(24)
We shall assume that the above problem is well-posed and endowed with a “sufficiently” regular solution u ∈
L2

(0, T ); Vγ

∩ C0

[0, T ]; L2(Ω)

. For example, by using similar arguments of [10], it is possible to show that
for ν = 1, V = 0, and σ(u(t)) = σ0 u(t) for some σ0 > 0, under the regularity hypothesis on the other data, there
exists a unique solution u ∈ L2

(0, T ); Vγ

∩ C0

[0, T ]; L2(Ω)

with ∂u
∂t ∈ L2

(0, T ); V′

, being V′ the dual space
of V. For further details see e.g. [37–40].
With the aim of writing problem (24) in the parameter space, we recall the mapping of the data based on Eqs. (1)
and (10). In particular, we have: 
V(ξ) := V(x(ξ)), 
f (ξ) := f (x(ξ)), 
γ (ξ) := γ (x(ξ)), and 
uin(ξ) := uin(x(ξ)); we
notice that 
σ(·) ≡ σ(·). Moreover, we introduce the affine manifold 
Vγ and the function space 
V as:

Vγ :=


v ∈ H1
(
Ω) : 
v|
ΓD
= 
γ

, 
V :=


v ∈ H1
(
Ω) : 
v|
ΓD
= 0

, (25)
where, if meas (Γ) ̸= 0, 
ΓD :=

ξ ∈ ∂
Ω : x(ξ) ∈ ΓD

and 
ΓN = ∂
Ω
ΓD. Also, we introduce the following forms
and functionals from Eqs. (10), (15) and (22):

a(
v, 
w) :=


Ω
µ 
∇
v ·


G−1 
∇
w


g d
Ω, 
b(
v, 
w) :=


Ω

v 
V ·


F 
G−1 
∇w


g d
Ω,

c(
w)(
v) :=


Ω

v
σ(
w)
g d
Ω, 
m(
v, 
w) :=


Ω

v 
w
g d
Ω,

q(
v) :=


Ω

v 
f 
g d
Ω,
(26)
and the weak residual:

Res (
w(t)) (
v) := 
q(
v) −
a(
v, 
w(t)) − 
b(
v, 
w(t)) −
c(
w(t))(
v) − ν 
m


v,
∂
w
∂t
(t)

, (27)
for any 
v ∈ 
V, 
w(t) ∈ 
Vγ , and for almost all t ∈ (0, T ). Finally, the weak form of the problem (24) in the parameter
space reads, for almost all t ∈ (0, T ):
find 
u(t) ∈ 
Vγ : 
Res (
u(t)) (
v) = 0 ∀
v ∈ 
V,
with 
u(0) = 
uin;
(28)
we observe that, from Eq. (10), we have u(x, t) = 
u(ξ, t) ◦ x−1(ξ) and 
u(ξ, t) = u(x(ξ), t) for almost all t ∈ (0, T ).
3.2. The Laplace–Beltrami equation
By referring to Eq. (20) and assuming the data and solution u as time independent, ν = 0, V = 0, and σ = 0, we
obtain the Laplace–Beltrami problem:
find u ∈ Vγ : a(v, u) = q(v) ∀v ∈ V, (29)
for some µ > 0. We observe that, if the manifold Ω is not endowed with a boundary, then the Laplace–Beltrami
problem (29) is ill-posed. For this reason, if meas (Γ) = 0, the solution space Vγ can be replaced e.g. by the space
V0 :=

v ∈ H1(Ω) :

Ω v dΩ = 0

(see e.g. [41]).
By recasting problem (29) in the parameter space, we obtain:
find 
u ∈ 
Vγ : 
a(
v,
u) = 
q(
v) ∀
v ∈ 
V, (30)
where we have used the notation introduced in Section 3.1.

9. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 815
3.3. The Laplace–Beltrami eigenvalue problem
From Eq. (20), by assuming that the data and solution u are time independent, and setting ν = 0, V = 0, and
σ(u) = −λ u, assuming the parameter λ as unknown, we obtain the Laplace–Beltrami eigenvalue problem5:
find u ∈ V and λ ∈ R : a(v, u) = λ m(v, u) ∀v ∈ V; (31)
specifically, we set µ = 1. We observe that, since the problem is symmetric, the eigenvalues are real valued, i.e. λ ∈ R,
with λ ≥ 0.
In the parameter space, Eq. (31) reads:
find 
u ∈ 
V and λ ∈ R : 
a(
v,
u) = λ 
m(
v,
u) ∀
v ∈ 
V. (32)
3.4. A time dependent linear advection–diffusion equation
By using the notation of Section 3.1, we introduce the parabolic linear advection–diffusion equation in the
following weak form, for almost all t ∈ (0, T ):
find u(t) ∈ Vγ : m

v,
∂u
∂t
(t)

+ a(v, u(t)) + b(v, u(t)) = q(v) ∀v ∈ V,
with u(0) = uin,
(33)
where we set σ(u(t)) = 0 and ν = 1. In view of the numerical approximation, we introduce the following operators,
say L(·) and Ladv(·), as:
L(w(t)) :=
∂w
∂t
(t) − µ ∆Ω w(t) + V · ∇Ω w(t), (34)
Ladv(w(t)) := V · ∇Ω w(t), (35)
and the strong residual R(·) in Ω as:
R (w(t)) := f − L(w(t)), (36)
for all w(t) ∈ H2(Ω) and for almost any t ∈ (0, T ).
In the parameter space, Eq. (33) reads, for almost all t ∈ (0, T ):
find 
u(t) ∈ 
Vγ : 
m


v,
∂
u
∂t
(t)

+
a(
v,
u(t)) + 
b(
v,
u(t)) = 
q(
v) ∀
v ∈ 
V,
with 
u(0) = 
uin,
(37)
while the operators and the residual in Eqs. (34)–(36) are:

L(
w(t)) :=
∂
w
∂t
(t) − µ
1

g

∇ ·


g 
G−1 
∇
w(t)

+ 
V ·


F 
G−1 
∇
w(t)

, (38)

Ladv(
w(t)) := 
V ·


F 
G−1 
∇
w(t)

, (39)

R (
w(t)) := 
f − 
L(
w(t)), (40)
for all 
w(t) ∈ H2(
Ω) and for almost any t ∈ (0, T ).
3.5. The Cahn–Allen equation
By denoting with u(t) the concentration of a component of a binary isothermal mixture, we describe its evolution
in time by means of the Cahn–Allen equation, a nonlinear, time dependent PDE. For further details on the topic we
refer the reader e.g. to [30,31,42,43]. The Cahn–Allen equation in weak form reads, for almost all t ∈ (0, T ):
5 For σ(u) = −λ u, the form c(·)(·) in Eq. (22) reads c(w)(v) = −λ m(v, u) for any v(x), w(t) ∈ V.

10. 816 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
find u(t) ∈ V : m

v,
∂u
∂t
(t)

+ a(v, u(t)) + c(u(t))(v) = 0 ∀v ∈ V,
with u(0) = uin,
(41)
where, with reference to Eq. (20), we choose σ(u(t)) = Ψc,u(u(t)) = 2u(t) (u(t) − 1) (2u(t) − 1) (for which we
have that c(u(t))(v) =

Ω v Ψc,u(u(t)) dΩ), V = 0, ν = 1, and f = 0. Moreover, if meas (Γ) ̸= 0, we set ΓN ≡ Γ
with ΓD = ∅. The Cahn–Allen equation (41) represents the gradient flow in the L2(Ω) norm of the following energy
functional (total free energy) for isothermal binary mixtures on surfaces:

Ψ(t) = Ψ(u(t)) :=

Ω

Ψc(u(t)) +
1
2
µ |∇Ω u(t)|2

dΩ, (42)
where Ψc(·) is the chemical energy, which, in this specific case, we have chosen as Ψc(u(t)) = (u(t))2 (u(t) − 1)2
;
the functional Ψc,u(·) = σ(·) represents the chemical potential and is obtained as the Frechét derivative of Ψc(·) with
respect to the variable u(t). The free energy 
Ψ(t) represents a Liapunov functional since d 
Ψ
dt (t) ≤ 0 for almost all
t ∈ (0, T ], being ∇Ω u(t) ·
nΓ = 0 on Γ.
In the parameter space, the Cahn–Allen equation reads, for almost all t ∈ (0, T ):
find 
u(t) ∈ 
V : 
m


v,
∂
u
∂t
(t)

+
a(
v,
u(t)) +
c(
u(t))(
v) = 0 ∀
v ∈ 
V,
with 
u(0) = 
uin,
(43)
with the free energy:

Ψ(t) = 
Ψ(u(t)) :=


Ω

Ψc(
u(t)) +
1
2
µ 
∇
u(t) ·


G−1 
∇
u(t)


g d
Ω, (44)
being 
Ψc(·) ≡ Ψc(·).
4. Numerical approximation
In this section we describe the numerical approximation of the PDEs introduced in Section 3.1 and represented in
compact form in Eq. (24). We discuss the approximation in a general setting, firstly for the spatial approximation by
means of Isogeometric Analysis and then, for time discretization, the generalized-α method.
4.1. The spatial approximation: isogeometric analysis
For the spatial approximation of the general problem (24) we consider Isogeometric Analysis. In this section we
briefly recall the basic notions of the approximation in the framework of the Galerkin method. For further details we
refer the reader to [17,18].
Isogeometric Analysis is a method for the spatial approximation of PDEs, based on the isoparametric concept for
which the same basis used to represent the known geometry are then used to approximate the unknown solution of
the PDEs. NURBS or B-splines geometries (computational domains) are represented by geometrical mappings in the
form (17) from a parameter space to the physical space. By using the isogeometric paradigm, we can define functions
in the parameter space in the form:

uh(ξ, t) :=
nbf

i=1

Ri (ξ) Ui (t), (45)
where 
Ri (ξ) are the NURBS (or B-splines) basis functions (see Eq. (18)) and Ui (t) ∈ R are the control variables (in
fact, the problem unknowns) for i = 1, . . . , nbf , with nbf the number of basis functions; for time dependent problems,
the control variables Ui (t) are time dependent. Since the geometrical mapping (17) is invertible (as discussed in
Sections 2.1 and 2.3), we can indifferently refer to functions in the parameter space or in the physical one; from

11. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 817
Eq. (10) we have uh(x, t) = 
uh(ξ, t) ◦ x−1(ξ). For this reason, unless than necessary for clarity, we will henceforth
consider formulations in the parameter space.
From Eq. (45) we define the NURBS space on the parameter domain 
Ω (a single patch):

Nh := span


Ri (ξ)
nb f
i=1
(46)
and, in virtue of the geometrical mapping (17), the corresponding NURBS space on the physical domain (the manifold)
Ω:
Nh := span


Ri (ξ) ◦ x−1
(ξ)
nb f
i=1
. (47)
Then, we introduce the finite dimensional function spaces 
Vγ,h := 
Vγ ∩ 
Nh and 
Vh := 
V ∩ 
Nh from Eqs. (25) and
(45); for simplicity, we assume that the data 
γ belongs to the NURBS space 
Nh. Similarly, by referring to the physical
domain Ω, we introduce the function spaces Vγ,h := Vγ ∩ Nh and Vh := V ∩ Nh.
The weak form of problem (28) approximated in space by Isogeometric Analysis reads, for almost all t ∈ (0, T ):
find 
uh(t) ∈ 
Vγ,h : 
Res (
uh(t)) (
vh) = 0 ∀
vh ∈ 
Vh,
with 
uh(0) = 
uin,h,
(48)
where 
uin,h is the L2(Ω) projection of 
uin onto 
Vh.6 The spatial approximation of the problems described in
Sections 3.2, 3.4 and 3.5 fit the general formulation of Eq. (48), while the eigenvalue problem (32) reads:
find 
uh ∈ 
Vh and λh ∈ R : 
a(
vh,
uh) = λh 
m(
vh,
uh) ∀
vh ∈ 
Vh. (49)
We remark that the function spaces 
Vγ,h and 
Vh can be “enriched” by means of h- or p-refinement of the geometric
representation while identically preserving the geometrical mapping.7 The refinements allow the improvement of
the accuracy of the approximate solution, while still representing exactly the NURBS geometries defining the
computational domain. A type of refinement called k-refinement and specific for NURBS basis functions can be
eventually used to efficiently improve the accuracy by increasing the degree p of the basis functions and their global
continuity, say k, across the elements 
Ωe in 
Ω while containing the number of basis functions nbf . For further details
we refer the reader to [17,18,20,21,45]. Specifically, in Section 5 we provide a priori error estimates for PDEs on
manifolds for h-refined meshes.
In some instances, as for example for closed surfaces, a sub-parametric Isogeometric approach can be considered,
for which the NURBS spaces 
Nh and Nh are built from different basis functions than those originally used in
the geometrical mapping x(ξ). This is the case when periodic NURBS basis functions are required for ensuring
global high order continuity for the NURBS spaces 
Nh and Nh. In particular, periodic NURBS basis functions are
obtained by means of piecewise (element by element) linear transformations of the original NURBS basis functions
and suitable master–slave constraints; see [26]. Namely, we should indicate with 
N
per
h := span


R
per
i (ξ)
n
per
bf
i=1 and
N
per
h := span


R
per
i (ξ) ◦ x−1(ξ)
n
per
bf
i=1 the periodic NURBS spaces with the periodic NURBS basis functions 
R
per
j (ξ)
obtained from 
Ri (ξ), for i = 1, . . . , nbf and j = 1, . . . , n
per
bf , even if, for notational simplicity, we will keep using
the notation of Eqs. (46) and (47); we remark that the geometrical mapping x(ξ) remains described in terms of the
original NURBS basis functions as in Eq. (17). In this manner, closed surfaces, as cylindrical shells, built on basis
functions which are only globally C0-continuous, may admit NURBS spaces of functions which are smoother and
eventually globally C p−1-continuous, a property which has revealed to be advantageous in several problems solved
by means of Isogeometric Analysis (see e.g. [45,46]).
4.2. The time discretization scheme
For the approximation of time dependent problems, we consider the generalized-α method, a predictor–
multicorrector numerical scheme, which we briefly recall in this section; for further details see also e.g. [32,47].
6 We remark that numerical integration is performed by means of the standard quadrature formula with (p + 1)κ Gauss–Legendre quadrature
points in each element; a direct method [44] is considered for the solution of the linear system associated to Eq. (48).
7 The h- or p-refinements correspond to the mesh refinement and order elevation procedures of the Finite Element or Spectral methods.

12. 818 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Let start by introducing a partition of the time interval (0, T ) into time steps {tn}nts
n=0, where t0 = 0 and
tnts = T , with time steps ∆tn := tn+1 − tn for n = 0, . . . , tnts−1. Also, we define U(t) := {Ui (t)}
nbf
i=1 , U̇(t) :=

U̇i (t)
nbf
i=1
, 
Res

U̇(t), U(t)

:=


Res (
uh(t))


Ri
nbf
i=1
from Eq. (27), Un := U (tn), and U̇n := U̇ (tn). By
introducing the parameters αm, α f , and δ ∈ R, the generalized-α method consists in solving the following problem at
the time step tn+1 given U̇n and Un:
find U̇n+1, Un+1, U̇n+αm , Un+α f : 
Res

U̇n+αm , Un+α f

= 0,
with: Un+1 = Un + ∆tn

(1 − δ) U̇n + δ U̇n+1

,
U̇n+αm = (1 − αm) U̇n + αm U̇n+1, Un+α f = (1 − α f ) Un + α f Un+1.
(50)
The previous problem is solved iteratively with Un+1 obtained as Un+1,( j) for some j = 0, . . . , jmax (with jmax > 1).
In particular, at the predictor stage ( j = 0), we set:
U̇n+1,(0) =
δ − 1
δ
U̇n, Un+1,(0) = Un. (51)
At the multicorrector stage, we repeat for j = 1, . . . , jmax the following steps:
1. evaluate the variables:
U̇n+αm,( j) = (1 − αm) U̇n + αm U̇n+1,( j−1),
Un+α f ,( j) = (1 − α f ) Un + α f Un+1,( j−1);
(52)
2. assemble the residual vector and tangent matrix:

Resn+1,( j) := 
Res

U̇n+αm,( j), Un+α f ,( j)

,

Kn+1,( j) := αm
∂ 
Res

U̇n+αm,( j), Un+α f ,( j)

∂U̇n+αm
+ α f δ ∆tn
∂ 
Res

U̇n+αm,( j), Un+α f ,( j)

∂Un+α f
;
(53)
3. if, for a prescribed tolerance tolR > 0, the criterion
∥
Resn+1,( j)∥
∥
Resn+1,(0)∥
≤ tolR is fulfilled, set U̇n+1 = U̇n+1,( j−1) and
Un+1 = Un+1,( j−1) and terminate the procedure, otherwise continue;
4. solve the linear system:

Kn+1,( j) ∆U̇n+1,( j) = −
Resn+1,( j); (54)
5. update the variables:
U̇n+1,( j) = U̇n+1,( j−1) + ∆U̇n+1,( j), Un+1,( j) = Un+1,( j−1) + δ ∆tn ∆U̇n+1,( j), (55)
and return to step 1.
A family of second-order time accurate and unconditionally stable generalized-α methods for linear problems is
obtained by choosing αm = 1
2

3−ρ∞
1+ρ∞

, α f = δ = 1
1+ρ∞
, where ρ∞ ∈ [0, 1] governs high frequencies
dissipation [9]8; a typical choice is ρ∞ = 0.5, see e.g. [24–26].
The time step ∆tn can be either fixed (∆tn = ∆t0) or determined by an adaptive algorithm. In the latter case, we
select an adaptive scheme based on the number of iterations of the multicorrector stage of the generalized-α method
(see e.g. [25,48]); in particular, based on numerical experience, we set ∆tn = ∆tn−1 χ

j
jref + j0
for n = 1, . . . , tnts−1
and an initial time step ∆t0, where χ = 1.2, jref = 2, and j0 = 0.8. As tolerance for the stopping criterion of the
multicorrector stage we select tolR = 10−4.
4.3. SUPG stabilization for the advection–diffusion PDEs
Since we are interested in solving time dependent advection–diffusion equations in the transport dominated regime,
we consider the numerical stabilization of the PDEs in lower dimensional manifolds. In particular, we consider a
8 The parameter ρ∞ represents the spectral radius of the amplification matrix of the system for ∆tn → ∞.

13. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 819
SUPG stabilization technique similar to the one proposed in [41] for PDEs on surfaces approximated by means of the
Finite Element method.
The time dependent linear advection–diffusion problem (37) with SUPG stabilization [33] reads, for almost all
t ∈ (0, T ):
find 
uh(t) ∈ 
Vγ,h : 
m


vh,
∂
uh
∂t
(t)

+
a(
vh,
uh(t)) + 
b(
vh,
uh(t))
+ 
dh(
vh,
uh(t)) − 
q(
vh) = 0 ∀
vh ∈ 
Vh,
with 
uh(0) = 
uin,h,
(56)
where the form 
dh(·, ·) represents the SUPG stabilization term. By recalling the definition (39) and (40), we define:

dh(
vh, 
wh(t)) :=
nel

e=1


Ωe
τe

Ladv (
vh) 
R(
wh(t))
g d
Ω, (57)
for all 
vh and 
wh(t) ∈ H2(
Ωe), with e = 1, . . . , nel, and t ∈ (0, T ); the stabilization parameter τe is assumed as
piecewise constant over the elements 
Ωe.9 We introduce the characteristic velocity Ve := ∥
V∥L∞(
Ωe), the constant
cp depending on the degree p of the polynomial approximation,10 and the local Péclet number Pee := Ve he
2µ .
Following [49–51], we choose:
τe :=
1
2

1
∆t2
+

Ve
he
2

1 +

cp
Pee
2
−1/2
, (58)
where he is the characteristic size (diameter) of the element Ωe (19) and ∆t is the time step for the time approximation;
we will assume cp = p2. This choice of τe is obtained by locally approximating the operator 
L−1(·), by assuming
the use of second order time schemes (as the generalized-α method), and by considering geometrical mappings of the
elements Ωe from parents domains (−1, 1)κ; the parameter τe assumes the same form in the parameter and physical
spaces.
5. A priori error estimation on lower dimensional manifolds
We provide the a priori error estimates under h-refinement for the Laplace–Beltrami equation (29) and the
Laplace–Beltrami eigenvalue problem (31) thus extending the results of [20] (and [21]) to the case of second order
PDEs defined on lower dimensional manifolds. With this aim, we propose some preliminary estimates for the change
of variables between the parameter and the physical domains and the interpolation error estimate for the NURBS
space on the physical domain (the lower dimensional manifold); for simplicity, we limit to consider estimates up to
the H1 norm. We remark that, for the derivation of the error estimate, we follow the procedure and recall the results
presented in [20].
Let us consider a family of “meshes”


Th

h
in the parameter domain 
Ω, with each mesh 
Th defined as in
Section 2.3. Following [20], we assume that such family of meshes is shape regular (i.e., for each mesh element

Ωe ∈ 
Th, the ratio between the smallest edge of the element and its diameter 
he is uniformly bounded with respect
to characteristic size of the mesh 
h); as consequence, the mesh 
Th is quasi-uniform (i.e. the ratio of the diameters of
any two neighboring elements is uniformly bounded with respect to 
h). We observe that, correspondingly to


Th

h
,
we have a family of meshes {Th}h on the manifold Ω obtained from the geometrical mapping (1). More general mesh
assumptions can be considered as e.g. in [52].
5.1. The interpolation error estimate
Let us recall the notation of Sections 2.1 and 2.3 regarding the geometrical mapping by NURBS and of
Section 4.1 for the NURBS spaces. Moreover, we recall the definition of the H1 semi-norm on the manifold
9 We notice that in the physical space we have, from Eqs. (35) and (36), dh(vh, wh(t)) :=
nel
e=1

Ωe
τe Ladv (vh) R(wh(t)) dΩ for all vh and
wh(t) ∈ H2(Ωe), with e = 1, . . . , nel, and t ∈ (0, T ).
10 The constant cp stems from an inverse inequality of the type ∥∆Ω vh∥L2(Ωe) ≤
cp
he
∥∇Ω vh∥L2(Ωe) for all vh ∈ Vh and e = 1, . . . , nel.

14. 820 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
|φ|H1(Ω) :=

Ω |∇Ω φ|2 dΩ
1/2
from Eq. (11); the definition of high order semi-norms follows consequently and
read |φ|Hm(Ω) :=

Ω |∇m
Ω φ|2 dΩ
1/2
for some m = 1, 2, . . . , being ∇m
Ω φ the vector of mth order derivatives of φ
on Ω. Similarly, the L2 norm on the manifold reads: ∥φ∥L2(Ω) :=

Ω |φ|2 dΩ
1/2
.
In analogy with [20] for the case κ = d, we provide the following preliminary results for functions defined on
manifolds represented by NURBS.
Proposition 5.1. Let us consider the general mesh element in the parameter domain 
Ωe ∈ 
Th and its corresponding
mesh element on the manifold Ωe ∈ Th. Then, for some integer m ∈ N, we obtain for φ ∈ Hm(Ωe) and 
φ ∈ Hm(
Ωe):
|
φ|Hm(
Ωe) ≤ Cshape




1
g




1/2
L∞(Ωe)
m

j=0
 

F

j
L∞(
Ωe)
|φ|H j (Ωe)

for m = 0, 1, 2 . . . , (59)
|φ|Hm(Ωe) ≤ C′
shape ∥
g∥
1/2
L∞(
Ωe)


F

−m
L∞(
Ωe)


φ


Hm(
Ωe)
for m = 0 or 1, (60)
where Cshape and C′
shape indicate positive and dimensionless constants depending on the shape of the manifold Ω.
Proof. Following [20] we introduce from Eqs. (1) and (2) the locally rescaled geometrical mapping:
x̆ : 
Ωe → Ω̆e, ξ → x̆(ξ), x̆(ξ) = x(ξ)


F

−1
L∞(
Ωe)
, (61)
for all 
Ωe ∈ 
Th, where Ω̆e belongs to the physical space. Correspondingly, we define its Jacobian:

F̆ : 
Ωe → Rd×κ
, ξ → 
F̆(ξ), 
F̆i,α(ξ) :=
∂x̆i
∂ξα
(ξ) i = 1, . . . , d, α = 1, . . . , κ, (62)
for all 
Ωe ∈ 
Th, and, in a similar manner, the high order differentials of the geometrical mapping 
F̆
(k)
: 
Ωe →
Rd×κ×···×κ for k ≥ 1 (see Eq. (3)); we remark that 
F̆ ≡ 
F̆
(1)
. We deduce that 
F(k) =


F


L∞(
Ωe)

F̆
(k)
for all
k = 1, . . . , m ≥ 1 and:



F(k)



L∞(
Ωe)
≤


F


L∞(
Ωe)





F̆
(k)




L∞(
Ωe)
for k = 1, . . . , m ≥ 1. (63)
By considering now the first order differential of the geometrical mapping (m = 1), it follows from Eqs. (2) and
(4) that 
F = 
F̆


F


L∞(
Ωe)
and 
G =

Ğ


F

2
L∞(
Ωe)
, where

Ğ :=


F̆
T 
F̆, from which we deduce that 
F 
G−1 =

F̆

Ğ
−1 

F

−1
L∞(
Ωe)
. Then, in analogy with Eqs. (6) and (8), we define F̆ : Ω̆e → Rd×κ with F̆(x) := 
F̆(ξ) ◦ x−1(ξ) and
Ğ : Ω̆e → Rκ×κ with Ğ(x) :=

Ğ(ξ) ◦ x−1(ξ), for all x ∈ Ω̆e; we obtain:


F G−1



L∞(Ωe)
≤


F̆ Ğ−1



L∞(Ω̆e)


F

−1
L∞(
Ωe)
. (64)
We start by proving the result (59) for m ≥ 1, the case m = 0 following similarly. From the definition of the
derivatives ∇m
Ω φ for m = 1, 2, . . . (see e.g. Eqs. (15) and (16)) and in analogy with [36], there exists a constant Cm
dependent only on m such that:


∇m
φ(ξ)

 ≤ Cm
m

j=1
|∇
j
Ω φ(x)|


i∈I ( j,m)


F(ξ)

i1



F(2)
(ξ)



i2
· · ·



F(m)
(ξ)



im

(65)
for all ξ ∈ 
Ωe with x = x(ξ) ∈ Ωe (in Euclidean norm), where 
∇m
φ is the mth order derivative of 
φ in 
Ω and
I ( j, m) :=

i = (i1, . . . , im) ∈ Nm :
m
l=1 il = j and
m
l=1 l il = m

. By recalling the definition of Hm semi-norm,
elevating the terms of the inequality to the square, integrating over the mesh element 
Ωe, and using an Hölder type
inequality, we obtain similarly to [20]:
|φ|Hm(
Ωe) ≤ Cm
m

j=1


Ωe


∇
j
Ω φ



2
d
Ωe
1/2


i∈I ( j,m)


F

i1
L∞(
Ωe)



F(2)



i2
L∞(
Ωe)
· · ·



F(m)



im
L∞(
Ωe)

. (66)

15. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 821
Since


Ωe


∇
j
Ω φ



2
d
Ωe =

Ωe


∇
j
Ω φ



2
1
g dΩe from Eq. (9), we obtain the following upper bound


Ωe

∇
j
Ω φ

2
d
Ωe
1/2
≤


1
g



1/2
L∞(Ωe)
|φ|H j (Ωe). From the definition of I ( j, m) and Eq. (63), we finally obtain the result (59) for m ≥ 1:


φ


Hm(
Ωe)
≤ Cm




1
g




1/2
L∞(Ωe)
m

j=1


F

j
L∞(
Ωe)
|φ|H j (Ωe)

·


i∈I ( j,m)




F̆



i1
L∞(
Ωe)





F̆
(2)




i2
L∞(
Ωe)
· · ·





F̆
(m)




im
L∞(
Ωe)

≤ Cshape




1
g




1/2
L∞(Ωe)
m

j=1


F

j
L∞(
Ωe)
|φ|H j (Ωe)

(67)
where Cshape depends on m, Cm, and




F̆



Wm,∞(
Ωe)
; we note that for m = 1 we can take Cshape = 1.
We proceed now to prove the result (60) in the case m = 1; the case m = 0 follows in a similar manner. From Eqs.
(6), (8) and (15) we have:
|∇Ω φ(x)| ≤


F(x) G−1
(x)





∇
φ(ξ)

 , (68)
for all x ∈ Ωe with ξ = x−1(ξ) ∈ 
Ωe (in Euclidean norm). We obtain that:
|φ|H1(Ωe) ≤


F G−1



L∞(Ωe)

Ωe


∇
φ

2
dΩe
1/2
(69)
and, since

Ωe


∇
φ

2
dΩe =


Ωe


∇
φ

2

g d
Ωe, we finally have:
|φ|H1(Ωe) ≤ ∥
g∥
1/2
L∞(
Ωe)


F G−1



L∞(Ωe)
|
φ|H1(
Ωe) = C′
shape ∥
g∥
1/2
L∞(
Ωe)


F

−1
L∞(
Ωe)
|
φ|H1(
Ωe), (70)
with C′
shape :=

F G−1


L∞(Ωe)


F


L∞(
Ωe)
. The choice of the constant C′
shape is justified by observing that from
Eq. (64) we have the bound C′
shape ≤


F̆ Ğ−1



L∞(Ω̆e)
and that, similarly to [20], F̆ is 0-homogeneous with respect to

F; hence, the constant C′
shape only depends on the shape of the manifold and is uniformly bounded with respect to the
mesh size he, since the NURBS mapping (1) is preserved under h-refinement.
We recall from [20] the interpolation error estimate for the NURBS space on the parametric domain, which is
obtained by using the concept of bent Sobolev space and combining Eqs. (42) and (65) reported in [20]. With this aim,
we introduce from Eq. (46) the projection operator Π 
Nh
: L2(
Ω) → 
Nh onto the NURBS space in the parameter
domain (see Eq. (41) in [20]).
Proposition 5.2. Given the indexes k and l such that 0 ≤ k ≤ l ≤ p + 1 and k ≤ 1, the mesh element 
Ωe ∈ 
Th, its
support extension 

Ωe, and the function 
φ ∈ Hl(

Ωe), we have:



φ − Π 
Nh

φ



Hk(
Ωe)
≤ Cshape

hl−k
e
l

i=0


Ω′
e∈ 
Th

Ω′
e∩

Ωe̸=∅


φ


Hi (
Ω′
e)
, (71)
where 
he is the characteristic size of the mesh element 
Ωe in the parameter domain and Cshape is a positive and
dimensionless constant depending on the NURBS parametrization but not on the size of 
Ωe.
Finally, we provide the interpolation error estimate for lower dimensional manifolds Ω defined by NURBS by
introducing from Eq. (47) the projection operator ΠNh
: L2(Ω) → 
Nh onto the NURBS space in the physical
domain, i.e. the “push-forward” of the NURBS projection operator, for which ΠNh
φ(x) =

Π 
Nh

φ(ξ)

◦ x−1(ξ) for
all φ ∈ L2(Ω) [20].

16. 822 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Theorem 5.1. Given the indexes k and l such that 0 ≤ k ≤ l ≤ p + 1 and k ≤ 1, the mesh element Ωe ∈ Th,
its support extension 
Ωe, and the function φ ∈ L2(Ω) ∩ Hl(
Ωe), the interpolation error estimate for the NURBS
space (47) on the lower dimensional manifold reads:

φ − ΠNh
φ


Hk(Ωe)
≤ Cshape hl−k
e
l

i=0


F

i−l
L∞(

Ωe)
|φ|Hi (
Ωe)

, (72)
where he is the characteristic size of the mesh element Ωe (19), 

Ωe is the support extension of the mesh element in
the parameter domain, and Cshape is a positive and dimensionless constant depending on the shape of the manifold
Ω but not its size.
Proof. We start by observing that from the definition of the NURBS projector ΠNh
and Eq. (60), we have:

φ − ΠNh
φ


Hk(Ωe)
≤ C′
shape ∥
g∥
1/2
L∞(
Ωe)


F

−k
L∞(
Ωe)



φ − Π 
Nh

φ



Hk(
Ωe)
, (73)
for k = 0 or 1. By using the result (71) of Proposition 5.2 for the term



φ − Π 
Nh

φ



Hk(
Ωe)
, we obtain:

φ − ΠNh
φ


Hk(Ωe)
≤ Cshape

hl−k
e ∥
g∥
1/2
L∞(
Ωe)


F

−k
L∞(
Ωe)
l

i=0


Ω′
e∈ 
Th

Ω′
e∩

Ωe̸=∅


φ


Hi (
Ω′
e)
, (74)
where the constants C′
shape and Cshape have been embedded and renominated in Cshape. We apply the result (59) of
Proposition 5.1 to each of the terms


φ


Hi (
Ω′
e)
which yields:


φ


Hi (
Ω′
e)
≤ Cshape




1
g




1/2
L∞(Ω′
e)
i

j=0
 

F

j
L∞(
Ω′
e)
|φ|H j (Ω′
e)

, (75)
for all 
Ω′
e ∈ 
Th, where Ω′
e is obtained from the geometrical mapping of 
Ω′
e. Then, by inserting Eq. (75) into Eq. (74),
and merging the double summation over the indexes i and j, for which 0 ≤ j ≤ i and 0 ≤ i ≤ l, we obtain:

φ − ΠNh
φ


Hk(Ωe)
≤ Cshape 
hl−k
e


F

−k
L∞(
Ωe)
l

i=0
 

F

i
L∞(

Ωe)
|φ|Hi (
Ωe)

, (76)
where we used the bound:


Ω′
e∈ 
Th

Ω′
e∩

Ωe̸=∅
 



1
g




1/2
L∞(Ω′
e)


F

i
L∞(
Ω′
e)
|φ|Hi (Ω′
e)

≤




1
g




1/2
L∞(
Ωe)


F

i
L∞(

Ωe)
|φ|Hi (
Ωe) , (77)
for all i = 0, . . . ,l, and the product ∥
g∥
1/2
L∞(
Ωe)


1
g



1/2
L∞(
Ωe)
has been embedded in the constant Cshape, similarly to
the proof of Proposition 5.1 and Theorem 3.1 in [20]. We recall the definition of characteristic mesh size he of the
element Ωe on the manifold given in Eq. (19) to obtain from Eq. (76):

φ − ΠNh
φ


Hk(Ωe)
≤ Cshape hl−k
e


F

−l
L∞(
Ωe)
l

i=0
 

F

i
L∞(

Ωe)
|φ|Hi (
Ωe)

. (78)
Finally, by multiplying and dividing the right hand side of Eq. (78) by


F

l
L∞(

Ωe)
and embedding the term


F

l
L∞(

Ωe)


F

−l
L∞(
Ωe)
into the constant Cshape we obtain the result (72).
In analogy with [20], a similar interpolation error estimation of Theorem 5.1 can be obtained for NURBS spaces
with boundary conditions. Moreover, we remark that the interpolation error estimate (72) for lower dimensional

17. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 823
manifolds assumes the same form of the result (61) of [20] in the case κ = d (Theorem 3.1), where the differences
between the two estimates have been embedded in the constant Cshape. When κ = d, the interpolation error
estimate (72) fully coincides with the estimate of Theorem 3.1 in [20].
5.2. The Laplace–Beltrami equation
By using the result of Theorem 5.1, we provide the a priori error estimates in L2 and H1 norms for the
Laplace–Beltrami problem of Section 3.2 defined on a general lower dimensional manifold Ω.
Following e.g. [8,53], by recalling the notation for the NURBS spaces introduced in Section 4.1 and assuming the
Laplace–Beltrami problem (29) to be well-posed, we have:
|u − uh|H1(Ω) ≤ C inf
vh∈Vγ,h
|u − vh|H1(Ω) , (79)
where the positive constant C depends on the data of the problem, specifically the coefficient µ (C = C(µ)). From
the result (72) of Theorem 5.1, by assuming quasi uniform h-refinement, the mesh elements of size he ≃ h for all
e = 1, . . . , nel, and the polynomial degree of the NURBS basis minα=1,...,κ pα = p, we obtain the error estimate in
semi-norm H1 for the case u ∈ H p+1(Ω):
|u − uh|H1(Ω) ≤ C hp
, (80)
where the positive constant C depends on µ, u, and the constant Cshape. Further, if we assume that homogeneous
Dirichlet boundary conditions (γ = 0) and u ∈ H p+1(Ω) ∩ H2(Ω) for all the data, we obtain from standard
Aubin–Nitsche arguments the error estimate in norm L2 (see e.g. [54]):
errL2 := ∥u − uh∥L2(Ω) ≤ C hp+1
, (81)
from which we deduce the error estimate in norm H1:
errH1 := ∥u − uh∥H1(Ω) =

err2
L2 + |u − uh|2
H1(Ω)
1/2
≤ C hp
. (82)
Similar considerations follow in the case for which the Laplace–Beltrami problem (29) is defined on lower
dimensional manifolds Ω not endowed with boundary; in this case, the analysis is performed by considering the
function space V0 in place of Vγ according to Section 3.2, which yields a similar result.
5.3. The Laplace–Beltrami eigenvalue problem
We provide the a priori error estimate for the numerical approximation of the Laplace–Beltrami eigenvalue problem
of Section 3.3 on the lower dimensional manifold Ω. We consider the error on the eigenvalues of the problem, say
|λn −λn,h|, where n indicates the nth eigenvalue and λn,h corresponds to its numerical approximation for n = 0, 1, . . .
(see e.g. Eq. (49)); specifically, we consider the ordering λ0 ≤ λ1 ≤ · · · ≤ λn ≤ · · · by recalling that the eigenvalues
of the Laplace–Beltrami eigenvalue problem (31) are real and non negative.
From [54] we have that, for the Laplace–Beltrami eigenvalue problem under consideration:
λn,h ≤
λn
1 − ϱn,h
∀n ≥ 0, (83)
where from Eq. (22):
ϱn,h := max
v∈Vn
∥v∥
L2(Ω)
=1


2 m

v − Π E
Vh
v, v

− m

v − Π E
Vh
v, v − Π E
Vh
v


 ∀n ≥ 0, (84)
with Vn ⊆ V the n-dimensional subspace spanned by the eigenfunctions u1, . . . , un and being Π E
Vh
: V → Vh
the Rayleigh–Ritz projector into the NURBS space on the physical domain (manifold) which yields, for some
φ ∈ V, a(vh, φ − Π E
Vh
φ) = 0 for all vh ∈ Vh. Since the NURBS space Nh is conforming with the function space V,

18. 824 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
by using the standard arguments presented in [54] (or [55]) and the interpolation error estimate (72) of Theorem 5.1,
we obtain:
λn ≤ λn,h ≤ λn + C (λn)p+1
h2p
∀n ≥ 0, (85)
provided that the mesh size h is sufficiently small (with he ≃ h for all the elements Ωe ∈ Th) and the polynomial
degrees of the NURBS basis such that minα=1,...,κ pα = p; the positive constant C depends on the data of the
Laplace–Beltrami operator and the constant Cshape. It follows that the error associated to the nth eigenvalue, say errn,
can be estimated as:
errn := λn,h − λn ∼ C (λn)p+1
h2p
∀n ≥ 0, (86)
which highlights the convergence rate 2p for h-refined mesh in agreement with the error estimates for the Finite
Element method in the case κ = d; see e.g. [53,54]. We remark that, when λn is “large” the error errn could be
significantly “large” also for fine meshes being the error proportional to (λn)p+1
.
6. Numerical results: PDEs on surfaces
We consider the numerical solution of the problems defined in Section 3 for the specific cases of surfaces as lower
dimensional manifolds.
6.1. The Laplace–Beltrami equation
We numerically solve the Laplace–Beltrami equation (29) on two surfaces, we evaluate the errors between the
numerical and exact solutions and we estimate the convergence orders of such errors for the polynomial degrees
p = 2 and 3 of the NURBS basis.
The Laplace–Beltrami problem on a cylindrical shell
Firstly, we consider as computational domain Ω a quarter of the cylinder represented in Fig. 3(left) (corresponding
to the quadrant with x ≥ 0 and y ≥ 0) with unitary radius and height L. We choose µ = 1 and f (φ, z) = β

α2π2
L2 gφ,1(φ) − gφ,2(φ)

gz(z), where φ := atan

x
y

, gφ,1(φ) := (1 − cos(φ)) (1 − sin(φ)) , gφ,2(φ) :=

cos(φ)+
sin(φ) − 4 sin(φ) cos(φ)

, and gz(z) := sin

α π z
L

for α ∈ N0 and β 0; then, we set u(x) = γ (x) = 0 on Γ.
The exact solution of the problem reads u(φ, z) = β gφ,1(φ) gz(z) in cylindrical coordinates. Specifically, we select
α = 3, β = 1/

3/2 −
√
2

, and L = 4. We solve the problem by meas of Isogeometric Analysis with NURBS
basis functions of degrees p = 2 and p = 3 for different mesh sizes starting from a mesh with 2 elements in the
circumferential direction. The exact and numerical solution and corresponding meshes are reported in Fig. 4; we
highlight the smoothness of the approximated solutions in circumferential direction even for a small number of mesh
elements. The convergence rates of the errors are reported in Fig. 5 for the polynomial degrees p = 2 and 3; in
particular, we obtain the convergence rates p + 1 and p for the norms L2 and H1, respectively. The convergence rates
are in agreement with the expected theoretical ones reported in Section 5.2 since the exact solution is “sufficiently”
regular, i.e. u ∈ C∞(Ω) ∩ H p+1(Ω).
The Laplace–Beltrami problem on a sphere
Then, we consider a Laplace–Beltrami problem with exact solution on the sphere of unitary radius reported in
Fig. 3(right); since the surface is closed (Γ ≡ ∅), we introduce a linear reaction term σ(u) = σ0 u (see Eq. (20)),
for which the problem is well-posed. We set the data σ0 = 1, µ = 1, and f (x) such that the exact solution is
u(x) = u(x, y, z) = x y2 − y z2 + x2 z, which does not exhibit any particular symmetry with respect to the poles of
the sphere (i.e. at z = ±1, being 0 the origin of the sphere of unitary radius).
For the numerical approximation we consider Isogeometric Analysis with the NURBS basis functions used for
the geometrical representation of the sphere, i.e. according to Section 2.3 by means of NURBS basis functions of
degree p ≥ 2, which are mostly C p−1-continuous on Ω except at the poles, the four meridians, and the equator of the
sphere, where they are only C0-continuous; for the sake of simplicity, we will refer to these NURBS basis functions as
C p−1/C0-continuous. We report in Fig. 6 the exact and numerical solutions corresponding to meshes progressively h-
refined; we remark the smoothness of the numerical solution along both the parametric directions even for the coarsest

19. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 825
Fig. 4. Laplace–Beltrami problem on a cylindrical shell. h-refined meshes (top), corresponding numerical solutions uh (bottom), and exact solution
u (bottom-right).
Fig. 5. Laplace–Beltrami problem on a cylindrical shell. Convergence of the errors errH1 (—) and errL2 (—◦) and reference convergences rates
p (- -) and p + 1 (. -) vs. the mesh size h for p = 2 (left) and p = 3 (right).
meshes. In Fig. 7 we highlight the convergence rates of the errors errL2 and errH1 which are in agreement with the
theoretical ones, being the exact solution u ∈ C∞(Ω) ∩ H p+1(Ω) as highlighted in Section 5.2.
As alternative to the use of NURBS basis functions which are C p−1/C0-continuous, we consider NURBS basis
functions which are periodic, i.e. in this case C p−1-continuous almost everywhere. As discussed in Section 4.1,
periodic basis functions can be generated by piecewise linear transformations of the original NURBS basis functions
used for the geometrical representation of the computational domain and the enforcement of master–slave constraints
(see also [26]). We can adopt this procedure for the NURBS basis functions used to geometrically represent the
sphere, with the aim of generating periodic NURBS basis functions which are “smother” on the sphere. However,
we remark that global C p−1-continuity can be achieved only in the parametric domain 
Ω, while, on the sphere Ω,
this property is achieved only almost everywhere, due to the singularities of the geometrical mapping at the poles.
Nevertheless, such periodic basis functions are globally C0-continuous and C p−1-continuous, except at the poles of

20. 826 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Fig. 6. Laplace–Beltrami problem on a sphere. h-refined meshes (top), corresponding numerical solutions uh (bottom) obtained by means of
Isogeometric Analysis with NURBS basis functions of degree p = 2 and C1/C0-continuous with number of elements of nonzero size nel = 32,
128, and 512, and exact solution u (bottom-right).
Fig. 7. Laplace–Beltrami problem on a sphere. Convergence of the errors errH1 (—) and errL2 (—◦) and reference convergences rates p (- -) and
p + 1 (. -) vs. the mesh size h obtained by means of Isogeometric Analysis with NURBS basis functions of degree p = 2, C1/C0-continuous (left)
and degree p = 3, C2/C0-continuous (right).
the sphere. We remark that, despite this case is namely outside the theory developed for the interpolation error estimate
in Theorem 5.1 (see e.g. Eq. (59)), we numerically verify the expected convergence rates for the errors in norms errL2
and errH1 under h-refinement, that is p + 1 and p, respectively; see e.g. Fig. 8(top) for the errors errL2 with p = 2
and 3.
We now compare the results obtained by means of NURBS-based Isogeometric Analysis with those obtained by
means of the isoparametric Finite Element method [9]. We remark that, in the latter case, piecewise Lagrangian
polynomial basis functions of degree p with global C0-continuity are used and that the surface of the sphere is
only approximated by means of finite elements. In Fig. 9 we compare the numerical results obtained by means of
NURBS-based Isogeometric Analysis with basis functions which are C p−1/C0-continuous and the isoparametric Fi-
nite Element method for the degrees p = 2 and 3 with nel = 8 mesh elements of nonzero size. As we can observe,

21. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 827
Fig. 8. Laplace–Beltrami problem on a sphere. Convergence of the errors errL2 and reference convergences rates p + 1 (. -) vs. the mesh size
h (top) and vs. the number of degrees of freedom nbf (bottom) obtained by means of Isogeometric Analysis with NURBS basis functions of
degree p which are C p−1/C0-continuous (—◦) and periodic (—×) and the isoparametric Finite Element method (—); degree p = 2 (left) and
p = 3 (right).
Fig. 9. Laplace–Beltrami problem on a sphere. Numerical solutions uh obtained by means of Isogeometric Analysis (IGA) and the isoparametric
Finite Element method (FEM) for the degrees p = 2 and p = 3 with a mesh comprised of nel = 8 elements of nonzero size; for Isogeometric
Analysis, NURBS basis functions of C1/C0- and C2/C0-continuities are considered for the degrees p = 2 and p = 3, respectively.
the geometrical approximation of the sphere associated to the Finite Element method is clearly visible, contrarily to
Isogeometric Analysis. Finally, in Fig. 8 we compare the numerical results obtained by means of Isogeometric Anal-
ysis with NURBS basis functions C p−1/C0-continuous and periodic with those associated to the isoparametric Finite

22. 828 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Fig. 10. Eigenvalue problem on a sphere. Eigenfunctions u4,1,h, u4,2,h, u5,1,h, and u6,1,h (from left to right) obtained by means of Isogeometric
Analysis with NURBS basis functions of degree p = 2 and C1/C0-continuous.
Fig. 11. Eigenvalue problem on a sphere. Convergence of the errors errn on the eigenvalues λn vs. the mesh size h for n = 4 (—), n = 5 (—◦),
and n = 6 (—) and reference convergences rate 2p (- -) for p = 2 (left) and p = 3 (right); the results have been obtained by means of Isogeometric
Analysis with NURBS basis functions which are C p−1/C0-continuous.
Element method; specifically, we compare the convergence rates of the errors errL2 vs. the mesh h, which are p + 1
for all the methods. However, we also notice, by plotting the errors errL2 vs. the number of degrees of freedom nbf ,
that the isoparametric Finite Element method is less efficient than its Isogeometric counterpart, since the achievement
of the same errors demands a much larger number of degrees of freedom. Finally, Fig. 8(bottom) indicates that, for
relatively coarse meshes, Isogeometric Analysis with smooth (periodic) basis functions provide more accurate results
than with the standard C p−1/C0-continuous basis functions.
6.2. The Laplace–Beltrami eigenvalue problem
We consider the numerical approximation of the eigenvalue problem (31) associated to the Laplace–Beltrami
operator on the sphere of unitary radius of Fig. 3(right). The exact values of the eigenvalues are λn = n(n + 1),
each with multiplicity 2n + 1, for n = 0, 1, . . . , ∞ (see e.g. [6]), to which correspond the eigenfunctions un,ln ,
with ln = 1, . . . , 2n + 1. We numerically approximate the problem by means of Isogeometric Analysis on a mesh
with nel = 131,072 elements and basis functions of degree p = 2 which are C1/C0-continuous; in Fig. 10 we
report for example the computed eigenfunctions u4,1,h, u4,2,h, u5,1,h, and u6,1,h corresponding to the eigenvalues
λ4 = 20, λ5 = 30, and λ6 = 42, respectively. In Fig. 11 we report the errors on the eigenvalues λ4, λ5, and
λ6 (errn = |λn − λn,h|), vs. the mesh size h when considering basis functions of degrees p = 2 and p = 3 (the
basis functions are C p−1/C0 continuous). As expected by the theoretical result (86) of Section 5.3, the convergence
rate obtained for the numerical results correspond to 2p; we remark that this result is obtained independently from the
fact that the eigenfunctions may or may not exhibit symmetries with respect to the poles, as highlighted for u5,1,h in
Fig. 10.

23. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 829
Fig. 12. Eigenvalue problem on a sphere. Normalized eigenvalue spectra, λn,h/λn vs. n/nbf , obtained by means of Isogeometric Analysis with
NURBS basis functions of degree p which are C p−1/C0-continuous (blue) and periodic (red) and the isoparametric Finite Element method (black)
for the degrees p = 2 (left) and p = 3 (right); the numbers of degrees of freedom are nbf = 8581, 8191, and 8065 for p = 2 and nbf = 9112, 8190,
and 18,240 for p = 3 obtained for the spatial approximations described above, respectively. (For interpretation of the references to color in this
figure legend, the reader is referred to the web version of this article.)
We now compare the normalized spectra obtained by considering the spatial approximation by means of
Isogeometric Analysis with the standard C p−1/C0-continuous basis functions and the periodic ones and by means
of the isoparametric Finite Element method. In Fig. 12 we plot λn,h/λn vs. n/nbf for the above mentioned
approximations by considering both the degrees p = 2 and p = 3; in the comparison, when possible, we
approximately use the same number of degrees of freedom nbf . As we can observe, the results obtained by means of
Isogeometric Analysis are significantly better than the ones obtained with the isoparametric Finite Element method for
the largest part of the spectrum, with evident advantages for n/nbf 
0.6. In addition, when considering Isogeometric
Analysis, the use of periodic (smooth) NURBS basis functions provides mode accurate results than their C p−1/C0-
continuous counterparts, both for p = 2 and 3; this improved accuracy advantage is clearly evident for n/nbf 
0.5.
6.3. The time dependent linear advection–diffusion equation
We consider the time dependent linear advection–diffusion equation on the cylindrical shell reported in Fig. 3(left).
By using the notation of Section 3.4 and using cylindrical coordinates, we set V(φ, z) = V0
√
1+a2
(
φ + a
z), and
f (φ, z) = e−b

(φ−φ0)2+(z−z0)2

, where φ := atan 2

x
y

, V0, a, φ0, and z0 ∈ R; we set homogeneous Neumann

24. 830 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Fig. 13. Time dependent advection–diffusion problem on a cylindrical shell. Evolution of the solution in time.
conditions on the boundary Γ = ΓN and uin(x) = 0. In particular, we set µ = 5.0 · 10−4, V0 = 1, a = 0.2,
b = 100, φ0 = π
4 , and z0 = 0.5. We numerically solve the problem by means of Isogeometric Analysis with the
SUPG stabilization technique discussed in Section 4.3. A NURBS representation with basis of degree p = 2 and
comprised of nel = 12,288 mesh elements is considered; the NURBS basis functions are C1/C0-continuous. For the
time discretization, the generalized-α method of Section 4.2 is used with the fixed time step ∆t0 = 1.5 · 10−2; we
choose the parameters αm = 5
6 and α f = δ = 2
3 corresponding to ρ∞ = 1
2 . In Fig. 13 we report the evolution of the
solution in time; we highlight its helical distribution along the surface of the cylindrical shell induced by the advective
field.
6.4. The Cahn–Allen problem
We solve the Cahn–Allen problem defined in Section 3.5 on the sphere of unitary radius of Fig. 3(right). By
using the notation of Section 3.5, we select µ = 5.0 · 10−4 and uin = u0 + ε, where u0 = 0.5 and ε is a random
distribution such that |ε| ≤ 0.1. We numerically solve the problem by means of Isogeometric Analysis with NURBS
basis functions of degree p = 2 and C1/C0-continuous for a mesh comprised of nel = 32,768 elements; the time
approximation is based on the generalized-α method with the adaptive time stepping scheme initialized with the time
step ∆t0 = 5.0 · 10−1 (as for the test of Section 6.3, we set αm = 5
6 and α f = δ = 2
3 ). In Fig. 14 we report the
evolution of the concentration u(x, t) in time. The phase transition evolves towards the steady state to a configuration
with the phases fully separated with a minimum perimeter interface. However, due to the fact that the Cahn–Allen
equation does not represent a mass conservative system, the solution evolves to a pure phase at the steady state, which
corresponds to u = 0 in the case under consideration. In Fig. 15 we report the evolution of the (Liapunov) free energy
functional 
Ψ(t) for which we observe that d 
Ψ
dt (t) ≤ 0.
In order to compare the results obtained with Isogeometric Analysis for NURBS basis functions featuring
different regularities and the isoparametric Finite Element method, we consider a Cahn–Allen problem with µ =
2.0 · 10−3, u0 = 0.5, and ε = 0.1 sin
√
17

x − 1
2

sin
√
18

y − 1
3

sin
√
19

z − 1
4

for the initial condition

25. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 831
Fig. 14. Cahn–Allen equation on a sphere. Evolution of the solution in time.
Fig. 15. Cahn–Allen equation on a sphere. Evolution of the free energy 
Ψ(t) in time.
uin = u0 + ε. In our comparison, we use Isogeometric Analysis with NURBS basis functions of degree p = 2
which are both C1/C0-continuous and periodic, as well as the isoparametric Finite Element method with Lagrangian
basis functions of degree 2; for these three cases, we use meshes comprised of nel = 8192, 8515, and 8192 elements
of nonzero size which yield the numbers of degrees of freedom nbf = 8582, 8192, and 32,514, respectively. In

26. 832 L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834
Fig. 16. Cahn–Allen equation on a sphere. Approximate solutions obtained at different times by Isogeometric Analysis (IGA) for C1/C0-
continuous basis functions with nbf = 8582 (a), IGA for periodic basis functions with nbf = 8192 (b), isoparametric Finite Elements (FEM)
with nbf = 32,514 (c), and FEM with nbf = 8192 (d).
addition, we consider for the isoparametric Finite Element method a case with a mesh of nel = 2048 elements and
nbf = 8066 in order to compare the results with Isogeometric Analysis by using approximatively the same number
of degrees of freedom rather than the same mesh size. A fixed time step ∆t = 1.0 · 10−1 is used for all the numerical
simulations, with the final time T = 500. In Fig. 16 we report the concentrations uh(x, t) obtained for the four spatial
discretizations discussed before at three significant time steps; we highlight that the result obtained with isoparametric
Finite Elements for nbf = 8192, i.e. for the “coarse” mesh, qualitatively differs from the other cases, a situation
which is mainly related to the geometrical approximation of the sphere. This is also confirmed in Fig. 17 by the
behavior of the (normalized) free energy functionals 
Ψ(t)/
Ψ(0) vs. the time t, for which significant discrepancies of
the energy are highlighted for the case of isoparametric Finite Elements with a relatively coarse mesh with respect to
Isogeometric Analysis approximations with approximately the same number of degrees of freedom.
7. Conclusions
In this work we showed the efficacy of Isogeometric Analysis for the numerical approximation of PDEs defined
on lower dimensional manifolds, specifically on surfaces. We considered different linear and nonlinear, elliptic and
parabolic PDEs with second order spatial operators of the Laplace–Beltrami type; examples include the eigenvalue
problem, the time dependent advection–diffusion equation, and the Cahn–Allen equation. We highlighted the
capability of Isogeometric Analysis of facilitating the encapsulation of the exact surface representations in the analysis
at their coarsest level of discretization, especially for geometries as cylindrical and spherical shells represented by
NURBS. Moreover, we provided a priori error estimates under h-refinement for the numerical approximation of

27. L. Dedè, A. Quarteroni / Comput. Methods Appl. Mech. Engrg. 284 (2015) 807–834 833
Fig. 17. Cahn–Allen equation on a sphere. Evolution of the normalized free energies 
Ψ(t)/
Ψ(0) in time (left) and zoom (right) obtained for
Isogeometric Analysis (IGA) and isoparametric Finite elements (FEM) spatial approximations of Fig. 16: IGA for C1/C0-continuous basis
functions with nbf = 8582 (blue), IGA for periodic basis functions with nbf = 8192 (black), FEM with nbf = 32, 514 (red), and FEM with
nbf = 8192 (green). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
second order PDEs on general lower dimensional manifolds in the case of linear and eigenvalues problems associated
to the Laplace–Beltrami operator. Specifically, the convergence rates of the errors are numerically confirmed for
benchmark tests cases which highlight the effect of the accurate geometrical description on the computation of the
numerical solutions. Moreover, we have shown that NURBS-based Isogeometric Analysis provides more accurate
results than isoparametric Finite Elements for PDEs defined on surfaces when considering the same number of degrees
of freedom.
Acknowledgments
We acknowledge Dmitry Alexeev for providing some preliminary insights into the topic and Andrea Bartezzaghi,
Dr. Claudia Colciago, Prof. John A. Evans, and Anna Tagliabue for the fruitful discussions.
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