A coupled bulk-surface finite element method is presented to solve problems arising in cell biology. Optimal order estimates for a linear elliptic equation are shown along with some numerical examples. An example of a parabolic problem with nonlinear coupling governed by Langmuir kinetics is presented, which describes the process of fluorescence recovery after photo bleaching (FRAP) in biological cells.

1. Applications of the Surface Finite Element
Method
Tom Ranner
Mathematics Department
September 2011
ENUMATH 2011, Leicester

2. The Problem
Many processes in biology and fluid mechanics are governed by
diffusion on a membrane or interface coupled to diffusion in an
enclosed bulk region. Other approaches to this problem include a
boundary integral formulation1 and a finite volume approach2 . We
wish to using the surface finite element method3 along with
standard finite element techniques4 for the bulk region.
1
Booty and Siegel 2010
2
Novak, Gao, Choi, Resasco, Schaff, and Slepchenko 2007
3
Dziuk 1988; Dziuk and Elliott 2007
4
Lenoir 1986

3. Example 1: Turing Instabilities
GTP-binding proteins (GTPase) molecules are important
regulators in cells that continuously run through an
activation/deactivation and
membrane-attachment/membrane-detachment cycle.
Activated GTPase is able to localise in parts of the
membranes and to induce cell polarity5 .
This can be modelled by Turing instabilities in a
reaction-diffusion system with attachment and detachment.
Figure 5
Rho GT Pases and cell migration. Cell migration requires actin-dependent protrusions at the front (re d)
and contractile actin:myosin filaments (re at the rear. In addition, microtubules (gre n) originating from
d) e
thecentrosome(purple arepreferentially stabilized in thedirection of migration allowing targeted vesicle
)
trafficking from the Golgi (b wn) to the leading edge.
ro
Figure: Example of cell polarisation from6
5
R¨tz and R¨ger 2011
a o
6
Jaffe and Hall 2005

4. Example 2: Surfactant problem
Surfactants, or surface contaminants, significantly alter the
interfacial properties of a fluid by changing the surface tension.
An example is the tip-streaming of thin threads or small droplets
from a drop or bubble this is stretched in an imposed extensional
flow7 .
FIG. 2. Microfluidic flow focusing geometry. ͑ a͒ Schematic diagram of the
design denoting flow of both the inner and outer liquids from left to right.
͑ b͒ Image of the orifice region of an actual microchannel ͓ outlined with the
dashed line in ͑ a͒ ͔ , including a typical image of the water-oil interface
extending toward the orifice from the upstream channel during a flow ex-
periment. Dimensions shown are Wup = 280 m, a = 90 m, ⌬Z = 180 m,
and Wor = 34 m.
8
Figure: Taken from
7
Booty and Siegel 2010
8
Anna and Mayer 2006

5. A Model Problem
As a model problem we will look to solve the following elliptic
problem:
We assume we have a bounded domain Ω ⊆ Rn with Lipschitz
boundary Γ. We wish to find u : Ω → R and v : Γ → R solution of
the system
−∆u + u = f in Ω (1a)
Γ
∂u Ω
(αu − βv ) + = 0 on Γ (1b)
∂ν
−∆Γ v + v − (αu − βv ) = g on Γ. (1c)
Here we assume α, β > 0 are given constants and f and g are
known functions on Ω and Γ respectively. We denote by ∆Γ the
Laplace-Beltrami operator on Γ.

6. A Model Problem: Weak form
We can convert this to a weak form using integration by parts over
Ω and Γ then using the boundary condition and taking an
appropriate linear combination of the two resulting forms leads to
the problem:
Find u : Ω → R and v : Γ → R such that
α u· η + uη + β Γv · Γξ + vξ
Ω Γ
+ (αu − βv )(αη − βξ) = α fη+β gξ
Γ Ω Γ
for all (η, ξ) ∈ H 1 (Ω) × H 1 (Γ). (2)

7. A Model Problem: Existence, uniqueness and regularity
Theorem (Existence and uniqueness)
If α, β > 0 and (f , g ) ∈ L2 (Ω) × L2 (Γ), there exists a unique pair
(u, v ) ∈ H 1 (Ω) × H 1 (Γ) which satisfies the weak form (2).
Proof.
Apply the Lax-Milgram theorem in the Hilbert space
H 1 (Ω) × H 1 (Γ) := {(η, ξ) : η ∈ H 1 (Ω) and ξ ∈ H 1 (Γ)},
with the norm
1
2 2 2
(η, ξ) H 1 (Ω)×H 1 (Γ) := η H 1 (Ω) + ξ H 1 (Γ)

8. A Model Problem: Existence, uniqueness and regularity
Theorem (Regularity)
Under the extra assumption that Γ ∈ C 2 then
(u, v ) ∈ H 2 (Ω) × H 2 (Γ) and
(u, v ) H 2 (Ω)×H 2 (Γ) ≤ c (f , g ) L2 (Ω)×L2 (Γ) .
Proof.
Apply standard regularity results for bulk domains9 and surface
domains10 to the weak form (2) with ξ = 0 and η = 0
respectively.
9
Gilbarg and Trudinger 1983
10
Aubin 1982

9. A Model Problem: Domain Discretisation
We define Ωh to be a polyhedral
approximation of Ω, with boundary
∂Ωh = Γh , such that nodes of Γh lie on Γ.
We take a quasi-uniform triangulation Th of
Γ
Ωh with simplices and define
h = max{diamT : T ∈ Th }.
h
Ωh
With this construction Th |Γh , the restriction
of Th to Γh is a quasi-uniform triangulation
of Γh . We assume that T ∩ Γh has at most
one face of T .

10. A Model Problem: Finite Element Approximation
We define the following finite element spaces:
Vh = {ηh : Ωh → R : ηh |T is linear, for each T ∈ Th }
Sh = {ξh : Γh → R : ξh |e is linear, for each e ∈ Th |Γh }.
The discrete problem is:
Find (uh , vh ) ∈ Vh × Sh such that
α uh · ηh + uh ηh + β Γh v h · Γh ξh + v h ξh
Ωh Γh
+ (αuh − βvh )(αηh − βξh ) = α fh ηh + β gh ξh
Γh Ωh Γh
for all (ηh , ξh ) ∈ Vh × Sh . (3)
fh and gh are some approximation of the data and will be specified
later.

11. A Model Problem: Numerical example
This method was implemented in the ALBERTA finite element
toolbox11 , with Ω = {(x, y , z) : x 2 + y 2 /2 + z 2 /3 < 1} with
α = β = 1, f (x, y , z) = 0 and g (x, y , z) = xy . fh and gh are taken
as the interpolants of f and g .
11
Schmidt, Siebert, K¨ster, and Heine 2005
o

12. A Model Problem: Numerical example
This method was implemented in the ALBERTA finite element
toolbox11 , with Ω = {(x, y , z) : x 2 + y 2 /2 + z 2 /3 < 1} with
α = β = 1, f (x, y , z) = 0 and g (x, y , z) = xy . fh and gh are taken
as the interpolants of f and g .
11
Schmidt, Siebert, K¨ster, and Heine 2005
o

13. A Model Problem: Abstract form
In order to perform error analysis, we introduce the following
abstract forms:
a(Ω) (w , η) = α w· η + wη
Ω
a(Γ) (y , ξ) = β Γy · Γξ + yξ
Γ
a(×) (w , y ), (η, ξ) = (αw − βy )(αη − βξ)
Γ
and
l (Ω) (η) = α fη l (Γ) (ξ) = β gξ
Ω Γ
finally,
a (w , y ), (η, ξ) = a(Ω) (w , η) + a(Γ) (y , ξ)
+ a(×) (w , y ), (η, ξ)
l (η, ξ) = l (Ω) (η) + l (Γ) (ξ).

14. A Model Problem: Abstract form
and the following approximate forms:
(Ω)
ah (wh , ηh ) = α wh · ηh + wh ηh
Ωh
(Γ)
ah (yh , ξh ) = β Γh yh · Γ h ξh + yh ξh
Γh
(×)
ah (wh , yh ), (ηh , ξh ) = (αwh − βyh )(αηh − βξh )
Γh
and
(Ω) (Γ)
lh (ηh ) = α fh ηh lh (ξh ) = β gh ξh .
Ωh Γh
finally,
(Ω) (Γh )
ah (wh , yh ), (ηh , ξh ) = ah (wh , ηh ) + ah (yh , ξh )
(×)
+ ah (wh , yh ), (ηh , ξh )
(Ω) (Γ)
lh (ηh , ξh ) = lh (ηh ) + lh (ξh ).

15. A Model Problem: Surface Lift
Since the exact problem and the discrete problem are posed on
different domains, we must relate the two domains. We start with
the surface.
We use normal projection of the
p(x) Γ d(x) ν(p(x)) domain. For h small enough, for
each point x ∈ Γh there exists a
x
x Γ
unique p(x) ∈ Γ. Given by
h
p(x) p(x) = x − d(x)ν(p(x)).

16. A Model Problem: Surface Geometric Estimates
These assumptions give us the following result12 .
Lemma
Let d denote a signed distance function for Γ, then
d L∞ (Γh ) ≤ ch2 .
If we denote by µh the quotient of the measures on the surface and
approximate surface, so that do = µh doh , we have that
sup |1 − µh | ≤ ch2 .
Γh
Let P denote projection onto the tangent space of Γ and Ph
projection onto the tangent space of Γh . We introduce the notation
1
Qh = µh (I − dH)PPh P(I − dH) then we have the estimate
|I − µh Qh | ≤ ch2 .
12
Dziuk 1988; Dziuk and Elliott 2007

17. A Model Problem: Bulk domain perturbation I
In order to relate the bulk domains, we introduce an exact
triangulation13 of Ω.
T
F
T
^
T
13
Bernardi 1989
14
Dubois 1987

18. A Model Problem: Bulk domain perturbation I
In order to relate the bulk domains, we introduce an exact
triangulation13 of Ω.
~
h
T T
~
F
T
^ F
T T
Dubois gives a construction of such maps FT to triangulate
smooth domains Ω14 , details of which will not be given here.
13
Bernardi 1989
14
Dubois 1987

19. A Model Problem: Bulk domain perturbation I
In order to relate the bulk domains, we introduce an exact
triangulation13 of Ω.
~
T G| hT T
~
F
T
^ F
T T
Dubois gives a construction of such maps FT to triangulate
smooth domains Ω14 , details of which will not be given here.
−1
We define Gh : Ωh → Ω locally by Gh |T : FT ◦ FT .
13
Bernardi 1989
14
Dubois 1987

20. A Model Problem: Bulk domain perturbation I
In order to relate the bulk domains, we introduce an exact
triangulation13 of Ω.
~
B h T B
h G| hT T
~
F
T
^ F
T T
Dubois gives a construction of such maps FT to triangulate
smooth domains Ω14 , details of which will not be given here.
−1
We define Gh : Ωh → Ω locally by Gh |T : FT ◦ FT .
This is a diffeomorphism and is the identity when restricted to
interior simplices, those with at most one boundary vertex.
We call the domain where Gh is different from the identity Bh .
13
Bernardi 1989
14
Dubois 1987

21. A Model Problem: Bulk domain perturbation II
Lemma (15 )
Let T ∈ Th be a boundary simplex and T the associated exact
triangle. We denote by Jh |T the absolute value of the determinant
of DGh |T . Under the assumption that Th is quasi-uniform and
Γ ∈ C 2 , then for sufficiently small h, we have that
DGh |T − I L∞ (T ) ≤ ch
D 2 Gh |T − I L∞ (T )
≤c
J h |T − I L∞ (T ) ≤ ch,
for some constant independent of T and h. Furthermore,
|Bh | ≤ ch2 ,
for some constant independent of h.
15
Lenoir 1986

22. A Model Problem: Lifted functions
For ηh ∈ Vh we define its lift ηh : Ω → R by
ηh (Gh (x)) := ηh (x).
For ξh ∈ Sh we define its lift ξh : Γ → R by
ξh (p(x)) := ξh (x).
We also define the lifted finite element functions
Vh = {ηh : ηh ∈ Vh } Sh = {ξh : ξh ∈ Sh }.
In the analysis, we will assume fh = f − and gh = g − to
avoid smoothness requirements.

23. A Model Problem: Error Bounds
Theorem
Let (u, v ) ∈ H 2 (Ω) × H 2 (Γ) be the solution of the variational
problem (2) and let (uh , vh ) ∈ Vh × Sh be the solution of the finite
element scheme given by (3). Denote by uh and vh the lifts of uh
and vh respectively. Then we have the following error bound:
(u − uh , v − vh ) ≤ C1 h
H 1 (Ω)×H 1 (Γ)
where
C1 = c (u, v ) H 2 (Ω)×H 2 (Γ) + (f , g ) L2 (Ω)×L2 (Γ) .

24. A Model Problem: Error Bounds (cont.)
Theorem (cont.)
Furthermore, if f ∈ L∞ (Ω) and u ∈ W 1,∞ (Ω) then
(u − uh , v − vh ) ≤ C2 h 2
L2 (Ω)×L2 (Γ)
where
C2 = c f L∞ (Ω) + g L2 (Γ) + u W 1,∞ (Ω) + u H 2 (Ω) + v H 2 (Γ) .

25. A Model Problem: Proof of error bounds I
Lemma (Approximation property16 )
For the lifted finite element spaces Vh and Sh , there exists an
interpolation operator Ih : H 2 (Ω) × H 2 (Γ) → Vh × Sh such that
w − Ih w L2 (Ω)×L2 (Γ) +h w − Ih w H 1 (Ω)×H 1 (Γ) ≤ ch2 w H 2 (Ω)×H 2 (Γ)
for all w ∈ H 2 (Ω) × H 2 (Γ).
16
Dziuk 1988; Lenoir 1986

26. A Model Problem: Proof of error bounds I
Lemma (Approximation property16 )
For the lifted finite element spaces Vh and Sh , there exists an
interpolation operator Ih : H 2 (Ω) × H 2 (Γ) → Vh × Sh such that
w − Ih w L2 (Ω)×L2 (Γ) +h w − Ih w H 1 (Ω)×H 1 (Γ) ≤ ch2 w H 2 (Ω)×H 2 (Γ)
for all w ∈ H 2 (Ω) × H 2 (Γ).
Lemma (Bulk domain errors)
Let wh , ηh ∈ Vh and denote their lifts by wh , ηh then
(Ω)
a(Ω) (wh , ηh ) − ah (wh , ηh ) ≤ ch wh ηh
H 1 (Ω) H 1 (Ω)
(Ω)
l (Ω) (ηh ) − lh (ηh ) ≤ ch f L2 (Ω) ηh .
L2 (Ω)
16
Dziuk 1988; Lenoir 1986

27. A Model Problem: Proof of error bounds II
Lemma (Surface domain errors)
Let (wh , yh ), (ηh , ξh ) ∈ Vh × Sh and let yh , ξh denote the lifts of
yh , ξh respectively and wh , ηh denote the lifts of the traces of
wh , ηh . Then
(Γ)
a(Γ) (yh , ξh ) − ah (yh , ξh )
≤ ch2 yh ξh H 1 (Γ)
H 1 (Γ)
(×)
a(×) (wh , yh ), (ηh , ξh ) − ah (wh , yh ), (ηh , ξh )
≤ ch2 (wh , yh ) (ηh , ξh )
L2 (Γ)×L2 (Γ) L2 (Γ)×L2 (Γ)
(Γ)
l (Γ) (ξh ) − lh (ξh )
≤ ch2 f L2 (Γ) ξh .
L2 (Γ)

28. A Model Problem: Domain approximation errors III
Lemma
Under the extra assumptions that f ∈ L∞ (Ω) and w ∈ W 1,∞ (Ω)
with w − the inverse lift of w onto Ωh . Let ηh ∈ Vh and denote its
lifts by ηh then
(Ω)
a(Ω) (w , ηh ) − ah (w − , ηh )
≤ ch2 w W 1,∞ (Ω) ηh H 1 (Ω)
(Ω)
l (Ω) (ηh ) − lh (ηh )
≤ ch2 f L∞ (Ω) ηh .
L2 (Ω)

29. A Model Problem: Proof H 1 error bound I
Notice for (ηh , ξh ) ∈ Vh × Sh , with lifts (ηh , ξh )
Fh (ηh , ξh ) := a (u − uh , v − vh ), (ηh , ξh )
= l (ηh , ξh ) − a (uh , vh ), (ηh , ξh )
= l (ηh , ξh ) − lh (ηh , ξh )
+ ah (uh , vh ), (ηh , ξh ) − a (uh , vh ), (ηh , ξh ).
Hence
Fh ηh , ξh ) ≤ C1 h ηh , ξh .
H 1 (Ω)×H 1 (Γ)

30. A Model Problem: Proof of H 1 error bound II
To prove the H 1 error bound, we rewrite the error as
a (u − uh , v − vh ), (u − uh , v − vh )
= a (u − uh , v − vh ), (u, v ) − Ih (u, v )
+ a (uh , v − vh ), Ih (u, v ) − (uh , vh )
= a (u − uh , v − vh ), (u, v ) − Ih (u, v )
+ Fh Ih (u, v ) − (uh , vh ) .
The result follows from the approximation property and the
domain error results.

31. A Model Problem: Proof L2 error bound I
To show the L2 bound we start by setting up a dual problem for
ζ ∈ L2 (Ω) × L2 (Γ):
Find wζ ∈ H 1 (Ω) × H 1 (Γ) such that
a (η, ξ), wζ ) = ζ, (η, ξ) L2 (Ω)×L2 (Γ) .
We assume this has a unique solution with the following regularity
result
wζ H 2 (Ω)×H 2 (Γ) ≤ c ζ L2 (Ω)×L2 (Γ) .

32. A Model Problem: Proof L2 error bound II
If we further assume that f ∈ L∞ (Ω) and u ∈ W 1,∞ (Ω), then we
achieve the improved bound
Fh (ηh , ξh ) ≤ C2 h2 (ηh , ξh ) .
H 1 (Ω)×H 1 (Γ)
This follows from applying the improved domain error results with
the bound |Bh | ≤ ch2 .

33. A Model Problem: Proof L2 error bound III
We start by writing the error
e = (u − uh , v − vh )
as the data for the dual problem and test with e so that
2
e L2 (Ω)×L2 (Γ) = a(e, we ).
Applying the interpolation theory, the H 1 bound, and the improved
bound on Fh leads to the L2 bound.

34. A Model Problem: Numerical Results I
To test the convergence rate, we applied this method with
α = β = 1 on the unit ball using the ALBERTA finite element
toolbox17 and the PARDISO numerical solver18 .
The data was chosen so that the exact solution was
u(x, y , z) = βxyz and v (x, y , z) = (3 + α)xyz.
We calculate the right hand side by setting fh and gh to be the
interpolants of f and g in Vh and Sh .
17
Schmidt, Siebert, K¨ster, and Heine 2005
o
18
Schenk, Waechter, and Hagemann 2007; Schenk, Bollhofer, and Roemer 2006

35. A Model Problem: Errors
h L2 error eoc H 1 error eoc
8.201523e-01 2.817753e-02 - 2.586266e-01 -
4.799888e-01 8.965317e-03 2.137583 1.517243e-01 0.995507
2.555341e-01 2.362206e-03 2.115725 7.961819e-02 1.022867
1.321787e-01 5.989949e-04 2.081457 4.024398e-02 1.035014
6.736035e-02 1.502987e-04 2.051078 2.016042e-02 1.025427
Table: Bulk errors: u − uh
h L2 error eoc H 1 error eoc
8.201523e-01 2.218139e-01 - 1.762399e+00
4.799888e-01 6.606584e-02 2.260828 9.810869e-01 1.093411
2.555341e-01 1.720835e-02 2.133950 5.028356e-01 1.060264
1.321787e-01 4.346577e-03 2.087386 2.529537e-01 1.042257
6.736035e-02 1.089298e-03 2.052899 1.266557e-01 1.026162
Table: Surface errors: v − vh

36. More realistic coupling – Langmuir Kinetics I
In many of the problems we consider, Langmuir kinetics govern the
coupling between bulk and surface concentrations. We consider a
bulk concentration u and a surface concentration v .
We assume the following simple boundary condition19
“The net flux across the interface is the net
absorbtion minus desorption rates.”
This is often interpreted by the adsorption-desroption flux on
the surface given by
κoff v − κon u(v − v ),
where κoff is a rate of desorption, κon is a rate of adsorption,
and v is a maximum desired surface concentration20 .
19
Kwon and Derby 2001
20
Georgievskii, Medvedev, and Stuchebrukhov 2002

37. More realistic coupling – Langmuir Kinetics II
Sometimes we wish to impose v as a maximum more strongly
by replacing the flux by
κoff v − κon u(v − v )+ ,
where ()+ denotes the positive part.
We can also consider the case where the surface concentration
is far from saturation
κoff v − κon u,
which is the linear case we have previously analysed.

38. More Realistic Coupling – Numerical Experiments
To test the different coupling models, we have implement the
following parabolic system. We wish to find u : Ω × [0, T ] → R
and v : Γ × [0, T ] → R such that
with q given by,
∂t u − ∆u = 0 q = 0,
∂u q = αu − βv ,
−q + =0
∂ν q = αu(1 − v ) − βv ,
∂t v − ∆Γ v + q = 0. q = αu(1 − v )+ − βv
We implemented this method using the above method to descritse
in space and semi-implicit time steping.

39. More Realistic Coupling – Numerical Experiments
Figure: From left to right: q = 0, q = αu − βv , q = αu(1 − v ) − βv ,
q = αu(1 − v )+ − βv

40. Turing Instabilities: Modelling
We look to model the concentrations of active and inactive
concentrations of GTPase on a cell membrane and in the
cystosol contained within21 .
We denote by Ω the cytosolic volume of the cell and Γ = ∂Ω
the cell membrane. We look for a bulk concentration
u : Ω → R of inactive GTPase, and surface concentrations,
v1 , v2 : Γ → R, of active and inactive GTPase, respectively.
We model the system using a
reaction-diffusion-attachment/detachment system.
For the reaction kinetics we assume simplified
Michaelis–Menten type law for catalysed reactions and a
Langmuir rate law for the attachment/detachment.
21
R¨tz and R¨ger 2011
a o

41. Turing Instabilities: Equations
Nondimensionalisation leads to the following system.
∂t u = D∆u in Ω
∂t v1 = d1 ∆Γ v1 + γf (v1 , v2 ) on Γ
∂t v2 = d2 ∆Γ v1 − γf (v1 , v2 ) + q(u, v1 , v2 ) on Γ
where
v1 v1
f (v1 , v2 ) = a1 + (a3 − a1 ) v2 − a4 ,
a2 + v1 a5 + v1
with the flux condition
−D u · ν = q(u, v1 , v2 ) on Γ
with
q(u, v1 , v2 ) = αu(1 − (v1 + v2 ))+ − βv2 .

42. Turing Instabilities: Weak Form
Using the same techniques as for the elliptic equation this leads to
the weak form:
Find (u, v1 , v2 ) ∈ H 1 (Ω) × H 1 (Γ) × H 1 (Γ) such that
α ∂t u η + D u · η +β ∂t v2 ξ + d2 Γ v2 · Γξ
Ω Γ
+ (αu(1 − (v1 + v2 ))+ − βv2 )(αη − βξ)
Γ
= −βγ f (v1 , v2 )ξ
Γ
∂t v1 ξ + d1 Γ v1 · Γ v2
Γ
=γ f (v1 , v2 )ξ,
Γ
for all (η, ξ) ∈ H 1 (Ω) × H 1 (Γ).

43. Turing Instabilities: Discretisation
We use a the above coupled bulk-surface finite element
method to discretize in space.
We discretise in time by treating the linear diffusion terms
implicitly and the nonlinear reaction and
attachment/detachment terms explicitly.
Parameter choices:
Diffusion Coefficients: D = 1000.0, d1 = 1.0, d2 = 1000.0
Reaction Coefficients: a1 = 0.0, a2 = 20.0, a3 = 160.0,
a4 = 1.0, a5 = 0.5
Nondimensionalised constant: γ = 400.0
Attachment/Detachment Coefficients: α = 0.1, β = 1.0
Discretisation: τ = 1.0e − 3, h = 0.137025.

44. Turing Instabilities: Numerical Results
Figure: Left: u, middle: v1 , right: v2

45. Turing Instabilities: Numerical Results
Figure: Left: u, middle: v1 , right: v2

46. Turing Instabilities: Numerical Results – Final Frame
Figure: Left: u (colour rescaled), middle: v1 , right: v2 , at final time

47. Conclusion
We have developed a computational method for solving
coupled bulk-surface partial differential equation.
We have analysed a model problem and derived optimal order
error estimates.
We have applied the method to a
reaction-diffusion-attachment/detachment problem from cell
biology.
In the future, we hope to perform analysis on the
reaction-diffusion-attachment/detachment problem and look
to derive error estimates.
We also hope to look at time dependent domains.

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Christine Bernardi. Optimal Finite-Element Interpolation on
Curved Domains. SIAM J. Numer. Anal., 26(5):1212–1240,
October 1989.
Michael R. Booty and Mchael Siegel. A hybrid numerical method
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Francois Dubois. Discrete vector potential representation of a
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