Abstract

List of publications

1 Introduction
1.1 Morphogenesis
1.2 Active matter
1.3 Surfaces and curvature in the context of morphogenesis
1.4 Challenges related to surfaces
1.5 State of the art of numerical methods for the solution of surface PDEs
1.5.1 Finite Element methods (FEMs)
1.5.2 Collocation methods
1.5.3 Other methods
1.6 This thesis: an overview

2 Surface differential operators
2.1 Motivation
2.2 Hierarchy of manifolds
2.3 Vector fields: covariant and contravariant notation
2.4 Differentiating vector fields on (curved) manifolds
2.4.1 Parallel transport
2.4.2 Covariant derivative
2.4.3 Covariant derivative in a Riemannian manifold
2.4.4 Summary
2.5 Covariant derivative of hypersurfaces in the Euclidean space
2.6 Curvature
2.6.1 Summary
2.7 A reference case: the S2 sphere
2.7.1 Operator expressions
2.7.2 Eigenfunctions of the scalar and vector Laplacian
2.7.3 An orthonormal basis on the S2 sphere

3 Surface DC-PSE 33
3.1 Particle methods for the approximation of differential operators
3.1.1 DC-PSE: Two-dimensional case
3.1.2 DC-PSE: General case
3.2 Approximating surface differential operators using particle methods
3.3 Surface DC-PSE
3.3.1 Neighborhood construction
3.3.2 Normal extension
3.3.3 Surface kernel construction
3.4 Verification and tests of Surface DC-PSE for scalar fields
3.4.1 Convergence tests
3.4.2 Analysis of the influence of rc and the curvature on the error
3.4.3 Strong scaling of Surface DC-PSE
3.4.4 Comparison with the Surface Finite Element Method
3.5 Reaction-diffusion and patterning with Surface DC-PSE

4 Vector surface PDEs
4.1 Surface DC-PSE for vector fields
4.2 Verification tests of Surface DC-PSE for vector fields
4.2.1 Convergence of vector Laplacian on the unit sphere
4.2.2 Convergence of vector diffusion on unit sphere
4.2.3 Vector diffusion on other shapes
4.3 Flows on surfaces
4.3.1 Verification of combined Surface DC-PSE and EDAC on the unit sphere 
4.3.2 Incompressible flow on tori of different curvature

5 Conclusion and Outlook
5.1 Summary
5.2 Limitations
5.3 Looking into the future

Appendices
A Numerical methods for function approximation and PDE solution
A.1 Collocation methods
A.2 Galerkin methods
A.3 Spectral methods
B Differential operators on the unit sphere S2
B.1 Gradient of a scalar
B.2 Gradient of a contravariant vector
B.3 Gradient of a covariant vector
B.4 Divergence of a contravariant vector
B.4.1 Divergence of a covariant vector
B.5 Laplacian (Laplace-Beltrami) of a scalar
B.6 (Connection) Laplacian of a contravariant vector
B.7 Crossproduct
C Scalar spherical harmonics
D Vector spherical harmonics
E An orthonormal basis on the unit S2 sphere
F DC-PSE algorithm: Auxiliary equations
G Simulation parameters
G.1 Fibonacci distribution of points on a sphere
G.2 Error metrics for vector quantities
G.3 Convergence of the Laplace-Beltrami operator on the unit circle and the unit
sphere
G.4 Mean and Gauss curvature of ellipsoid and Stanford bunny
G.5 Analysis of the influence of rc and the curvature on the error of the Laplace-Beltrami operator on the unit sphere
G.6 Comparison with FEMs
G.7 Reaction-diffusion patterns
G.8 Convergence of the connection Laplacian on the unit sphere
G.9 Convergence of the surface divergence operator on the unit sphere
G.10 Convergence of the solution of the vector diffusion equation on the unit sphere G.11 Vector diffusion equation on the torus and a nonic surface
G.12 Verification of combined Surface DC-PSE and EDAC for the solution of the
incompressible surface Navier-Stokes equation on the unit sphere 
G.13 Test for the incompressible surface Navier-Stokes equation on tori
G.14 Spherical harmonics for the solution of surface diffusion on the unit sphere
G.15 Error computation using Richardson extrapolation

Bibliography

Index

Acknowledgments