Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.

1. How to Solve a Partial Differential Equation on a
Surface
Tom Ranner
University of Warwick
T.Ranner@warwick.ac.uk
http://go.warwick.ac.uk/tranner
University of Warwick, Graduate Seminar,
3rd November 2010

2. Where do surface partial differential equations come from?
Partial differential equations on surfaces occur naturally in many
different applications for example:
fluid dynamics,
materials science,
cell biology,
mathematical imaging,
several others. . .

3. Example Surfaces – Cell Biology
Figure: An endoplasmic reticulum (ER)

4. Example Surfaces – Pattern Formation
Invited Article R253
Figure 1. Examples of self-organization in biology. Clockwise from top-left: feather bud
patterning, somite formation, jaguar coat markings, digit patterning. The first is reprinted from
Widelitz et al 2000, β-catenin in epithelial morphogenesis: conversion of part of avian foot scales
into feather buds with a mutated β-catenin Dev. Biol. 219 98–114, with permission from Elsevier.
The second is courtesy of the Pourqui´ Laboratory, Stowers Institute for Medical Research, and
e
the remainder are taken from the public image reference library at http://www.morguefile.com/.
(a) Activator (b) Inhibitor
80 80 8
15
Taken from R E Baker, E A Gaffney and P K Maini, Partial differential equations for self-organisation in cellular
60 60
Distance (y)
Distance (y)
and development biology. Nonlinearity 21 (2008) R251–R290.
40 10 40 4
20 20

5. Example Surfaces – Dealloyed surface
The surface can be the interface of two materials in an alloy.
9738 C. Eilks, C.M. Elliott / Journal of Computational Physics 227 (2008) 9727–9741
Fig. 3. Simulation on a large square, t ¼ 0:04; t ¼ 0:1 and t ¼ 0:2.
Taken from C The geometric motion ofNumerical simulation of dealloying by surface nonvanishing right hand evolving surface
plane. Eilks, C M Elliott, the surface has little influence, except for providing for a dissolution via the side for the
conservation law of gold on the surface, since gold from the bulk is accumulating on the surface. While the concrete
finite element method. Journal of Computational Physics 227 random distribution in the bulk, the lengthscales of the
appearance of the structure obviously depends on the particular (2008) 9727–9741.
structure depend only on the particular values of the parameters in the equation.
After the phases have separated, etching still continues in the areas with a small concentration of gold, while the motion
is negligible in regions covered by gold yielding a maze like structure of the surface. The origins of this shape can still be
explained by the initial phase separation which fixed the gold covered regions that proceeded to move into the bulk. So
at this stage the simulation does not necessarily show the mechanisms for the emergence of a nanoporous structure. By
undercutting the gold rich portion of the surface, the area of the surface that is not covered by gold increases. In the last
stages of this simulation new components of the gold rich phase emerge at the bottom of the surface. Additionally the inter-
face separating gold rich and gold poor phases shows no effect of coarsening as for the planar Cahn-Hilliard equation, but
instead becomes more complicated. These two effects can be seen as signs that the model shows increasing formation of
morphological complexity. We explore them in more detail in the following examples. Note however that due to self-inter-
sections the surface is not embedded at later stages, as can be seen in Fig. 4, where the cross sections along the plane parallel

6. 1D Heat Equation
The one dimensional heat equation is given by
ut = uxx on (0, 1) × (0, T )
u(0, t) = u(1, t) = 0 for t ∈ (0, T )
u(x, t) = u0 (x) for x ∈ (0, 1).
Joseph Fourier solved this in 1822 using separation of variables.
The idea is to write
u(x, t) = X (x)T (t)
and derive a system of decoupled odes for X and T , which can
then be solved simply.

7. 2D Heat Equations
In two dimensions the heat equation on a square
Ω = (0, 1) × (0, 1) becomes
ut = ∆u := uxx + uyy in Ω × (0, T )
u = 0 on ∂Ω × [0, T ]
u(x, 0) = u0 (x) on Ω × {0}.
This can be solved using Fourier analysis. We rewrite
u(x, t) = uk (t)e ik·x ,
ˆ
k∈Z2
and derive a system of odes of uk .
ˆ

8. What is the surface Heat Equation?
For a d-dimensional hypersurface Γ we would like to write down
something like ut = ∆u but u is only define on Γ. So instead we
have
ut = ∆Γ u,
where ∆Γ is the Laplace-Beltrami operator.

9. Surface gradients
For a function η : Γ → R we define the surface gradient Γη
of η by
Γη = η − (ν · η)
where ν is the outward pointing normal on Γ and η is the
gradient in ambient coordinates.

10. Surface gradients
For a function η : Γ → R we define the surface gradient Γη
of η by
Γη = η − (ν · η)
where ν is the outward pointing normal on Γ and η is the
gradient in ambient coordinates.
The Laplace-Beltrami operator ∆Γ is given by the surface
divergence of the surface gradient
∆Γ η = Γ · Γ η.

11. Surface Heat Equation
The surface heat equation on a closed surface Γ is given by
ut = ∆Γ u on Γ × [0, T ]
u(x, 0) = u(x) on Γ.

12. Surface Heat Equation
The surface heat equation on a closed surface Γ is given by
ut = ∆Γ u on Γ × [0, T ]
u(x, 0) = u(x) on Γ.
How do we solve this equation?
separation of variables
Fourier analysis
parametrisation
approximation. . .

13. Approximation of the 1D Heat Equation – using finite
differences
Let’s first go back to the 1D problem is demonstrate some possible
methods. Take the 1D heat equation
ut = uxx ,
and let’s approximate the derivatives by finite differences. We
divide (0, 1) into N intervals of length ∆x, (xj , xj+1 ) for
j = 0, . . . N. Our approximate solution u h solves for
j = 1, . . . , N − 1 and t ∈ (0, T )
h u h (xj−1 , t) − 2u h (xj , t) + u h (xj+1 , t)
ut (xj , t) = .
∆x 2

14. Approximation of the 1D Heat Equation – using finite
differences
From, for j = 1, . . . , N − 1 and t ∈ (0, T )
h u h (xj−1 , t) − 2u h (xj , t) + u h (xj+1 , t)
ut (xj , t) = ,
∆x 2
we wish to discretize in time. We approximate the time derivative
on the left hand side in the same way, but there is a choice on the
right-hand side to evaluate at t = tk or tk+1 . We choose for
numerical reasons the Backwards Euler method t = tk+1 . So we
have a linear system
u h (xj , tk+1 ) − u h (xj , tk ) u h (xj−1 , tk+1 ) − 2u h (xj , tk+1 ) + u h (xj+1 , tk+1
=
∆t ∆x 2

15. Approximation of the 1D Heat Equation – using finite
differences
The solve strategy is then
1. Initialise u h (xj , 0) = u0 (xj ) for j = 0, . . . , N
2. For k = 0, 1, 2, . . . solve the linear system
∆t h
u h (xj , tk+1 ) − u (xj−1 , tk+1 ) − 2u h (xj , tk+1 ) + u h (xj+1 , tk+1 )
∆x 2
= u h (xj , tk )
for j = 1, . . . , N − 1, and
u h (xj , tk+1 ) = 0
for j = 0, N.

16. Approximation of the 1D Heat Equation – using finite
differences
This was implemented in MATLAB with the following result:

17. Approximation of the 2D Heat Equation – using finite
differences
The same method can be implemented for the 2D problem, with
the following result:

18. Can finite differences work on a surface?
This method works best on a regular grid, which is almost
always impossible on a surface.
One must parameterise the surface first!
Projections or embeddings can be used to solve a the surface
pde using this type of method. An example of this type of
method is the closest point method.
Instead, we can try to approximate the solution rather than
the problem.

19. Approximation of the 1D Heat Equation – using finite
elements
Again we split the domain (0, 1) into N intervals of length h. We
define the space Vh of finite element functions to be
Vh = {ηh ∈ C0 (0, 1) : ηh |(xj ,xj+1 ) is affine linear for each j}.
We would like to solve
h h
ut = uxx
so that u h (·, t) ∈ Vh for all t, but the finite element functions
don’t have two derivatives in space!

20. Approximation of the 1D Heat Equation – using finite
elements
The finite element functions do have a first derivative almost
everywhere, so we put the equations in integral form to remove
one of the derivatives.

21. Approximation of the 1D Heat Equation – using finite
elements
The finite element functions do have a first derivative almost
everywhere, so we put the equations in integral form to remove
one of the derivatives. We multiply by a test function φ and
integrate with respect to x
1 1
ut φ = uxx φ,
0 0
we then integrate by parts using the boundary condition
u(0) = u(1) = 0 to get
1 1
ut φ + ux φ x = 0
0 0
Now all the terms in the above equation exist for all
1
u, φ ∈ Vh ⊆ H0 (0, 1). This is called the weak form of the heat
equation.

22. Approximation of the 1D Heat Equation – using finite
elements
We wish to find u h (·, t) ∈ Vh such that for all time t
1 1
h h
ut φ + ux φx = 0.
0 0
We would like this to be true for all φ ∈ Vh , but this is
equivalent to being true for all basis functions φj ∈ Vh .

23. Approximation of the 1D Heat Equation – using finite
elements
We wish to find u h (·, t) ∈ Vh such that for all time t
1 1
h h
ut φ + ux φx = 0.
0 0
We would like this to be true for all φ ∈ Vh , but this is
equivalent to being true for all basis functions φj ∈ Vh .
We can find a basis for Vh by setting φj (xi ) = δij . Our
problem is to find u h (·, t) ∈ Vh for t such that
1 1
h h
ut φ j + ux (φj )x = 0 for j = 1, . . . , N.
0 0
Notice that the boundary condition is automatically satisfied if
u h ∈ Vh .

24. Approximation of the 1D Heat Equation – using finite
elements
We can decompose u h in terms of the basis function φj to get
N
u h (x, t) = Ui (t)φj (x).
i=0
The equations become
1 1
Ui,t φi φj + Ui (φi )x (φj )x = 0,
0 i 0 i
If we write U(t) = (U0 (t), . . . , UN (t)) and define the mass matrix
M and stiffness matrix S by
1 1
Mij = φi φj Sij (φi )x (φj )x ,
0 0
we can write a matrix system
MUt = SU.

25. Approximation of the 1D Heat Equation – using finite
elements
We discretize in time using backwards Euler again to get the
following solve strategy:
1. Initialise U 0 as Uj0 = u0 (xj ).
2. For each k = 0, . . ., solve the linear system
(M + ∆tS)U k+1 = MU k .

26. Approximation of the 1D Heat Equation – using finite
elements
This method was implemented in MATLAB with the following
result:

27. Approximation of the 2D Heat Equation – using finite
elements
This method can also be used for the 2D problem:

28. tinuously; the method should also be robust enough to tolerate different resolutions
and boundaries; for database indexing, each class index should be small for storage
Can we use finite elements for surface partial differential
and easy to compute.
Conformal mapping has many nice properties to make it suitable for classification
problems. Conformal mapping only depends on the Riemann metric and is indepen-
equations? dent of triangulation. Conformal mapping is continuously dependent of Riemann
metric, so it works well for different resolutions. Conformal invariants can be repre-
sented as a complex matrix, which can be easily stored and compared. We propose to
use conformal structures to classify general surfaces. For each conformally equivalent
class, we can define canonical parametrization for the purpose of comparison.
Geometry matching can be formulated to find an isometry between two surfaces.
We can use what is called a surface finite element method, in
By computing conformal parametrization, the isometry can be obtained easily. For
which we approximate the domain Γ also.
surfaces with close metrics, conformal parametrization can also give the best geometric
matching results.
Fig. 1. Surface & mesh with 20000 faces
1.1. Preliminaries. In this section, we give a brief summary of concepts and
Justification of this method including well posedness, stability and convergence can be found in G. Dziuk and C. M.
notations.
Elliott, Surface finite elements for parabolic equations. J.realization |K| is homeomorphic
Let K be a simplicial complex whose topological Comput. Math. 25(4), (2007) 385–407. Image taken
from Xianfeng Gucompact 2-dimensional manifold. SupposeConformal Structures of surfaces. Communications in
to a and Shing-Tung Yau Computing there is a piecewise linear embedding
Information and Systems. 2(2) (2002), 121–146.
(1) F : |K| → R3 .
The pair (K, F ) is called a triangular mesh and we denote it as M . The q-cells of K

29. Solving the surface heat equation – using surface finite
elements
We define Γh to be a polyhedral approximation of Γ (made of
triangles) with vertices xi .
Vh is the space of piecewise linear functions on Γh with basis
φj given by φj (xi ) = δij
We look to solve the weak form of the surface heat equation
on Γh :
h h
ut φ + Γh u · Γh φ = 0.
Γh Γh

30. Solving the surface heat equation – using surface finite
elements
We define the surface mass matrix M and surface stiffness
matrix S by
Mij = φi φj Sij = Γ h φi · Γ h φj
Γh Γh
Using the same notation for U as before we have
MUt + SU = 0.

31. Solving the surface heat equation – using surface finite
elements
We discretize using backwards Euler to get the solve strategy
1. Initialise U 0 by Uj0 = u0 (xj )
2. For k = 0, 1, . . ., solve the linear system
(M + ∆tS)U k+1 = MU k .

32. Solving the surface heat equation – using surface finite
elements
This method was implemented using the ALBERTA finite element
toolbox on S 2 to get:

33. Optimal Partition Problem
Given a surface Γ, a positive integer m and parameter ε > 0, we
wish to minimised the following energy functional for a
vector-valued function u = {uj } ∈ (H 1 (Γ))m :
2
E ε (u) = (| Γ u| + 2Fε (u)),
Γ
where Fε is in interaction potential of the form
m
1
Fε (u) = 2 ui2 + uj2 .
ε
i=1 j<i
Based on Quang Du and Fangda Lin, Numerical approximations of a norm-preserving gradinet flow and
applications to an optimal partition problem Nonlinearity 22 (2009) 67–83.

34. Optimal Partition Problem – Gradient Decsent
If we minimised E ε subject to uj L2 (Γ) = 1 for each j we have the
following norm preserving gradient decent equations:
uj,t − ∆Γ uj = λj (t)uj − Fε,uj (u) on Γ × R+
|uj |2 = 1 for j = 1, . . . , m.
Γ
along with the initial condition
u(0, x) = g (x) ∈ H 1 (Γ, Ξ), with gj L2 (Γ) = 1.
Here Ξ is the singular subset in Rm
 
 m 
Ξ = y ∈ Rm : yj2 yk = 0, yk ≥ 0 for 1 ≤ k ≤ m .
2
 
j=1 k<j

35. Optimal Partition Problem – Partition?
Since the vector-valued function u takes values on in Ξ, the above
system is equivalent to the following problem
For a given surface Γ, partition Γ into m region Γj such
that the sum j λj , with λj the first eigenvalue of ∆Γ
over Γj with a Dirichlet boundary condition, minimised.

36. Optimal Partition Problem – How to solve?
To progress from step tn to tn+1 , given u n we calculate u n+1 by
1. Set u n+1 as the solution the heat equation at t = tn+1
ˆ
ut = ∆Γ u for x ∈ Γ, tn < t < tn+1
u(tn , x) = u n (x).
2. use (Gauss-Seidel) solver of decoupled ODEs: sequentially for
j = 1. . . . , m,
 
2uj 
uj,t = − 2 (˜in+1 )2 +
u (ˆin+1 )2  for x ∈ Γ, tn < t < tn+1
u
ε
i<j i>j
uj (tn , x) = ujn+1 (x)
ˆ
Set u n+1 as the solution u(tn+1 , x) of this system at t = tn+1 ,
˜
3. normalisation: we set u n+1 via
ujn+1
˜
ujn+1 = for j = 1, . . . , m,
ujn+1
˜
L2 (Γ)

37. Optimal Partition Problem – Numerical Results
This equation was discretize using a surface finite element method
and solved here for m = 6 on S 2 .

38. Summary
Methods such as Fourier series and separation of variables
don’t work on general surface so we must approximate
solutions.
Finite difference methods work by approximating the
differential operator by difference quotients, but only work
best on rectangular domains.
Finite element methods approximate weak solutions by
dividing up the domain into finitely many subdomains and can
be extended to surface problems.
A surface finite element method can be used to solve many
different types of problems on surfaces.