## Lectures

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### 2006/12/08

10:30-12:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Computational Methods for Surface Partial Differential Equations

**Charles M. Elliott**(University of Sussex)Computational Methods for Surface Partial Differential Equations

[ Abstract ]

In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms

of elliptic operators which naturally generate weak formulations

of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).

In these lectures we discuss the formulation, approximation and applications of partial differential equations on stationary and evolving surfaces. Partial differential equations on surfaces occur in many applications. For example, traditionally they arise naturally in fluid dynamics, materials science, pattern formation on biological organisms and more recently in the mathematics of images. We will derive the conservation law on evolving surfaces and formulate a number of equations.

We propose a surface finite element method (SFEM) for the numerical solution of parabolic partial differential equations on hypersurfaces $\\Gamma$ in $\\mathbb R^{n+1}$. The key idea is based on the approximation of $\\Gamma$ by a polyhedral surface $\\Gamma_h$ consisting of a union of simplices (triangles for $n=2$, intervals for $n=1$) with vertices on $\\Gamma$. A finite element space of functions is then defined by taking the continuous functions on $\\Gamma_h$ which are linear affine on each simplex of the polygonal surface. We use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on $\\Gamma$. Our finite element method is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. We give an example of error bounds in the case of semi-discretization in space for a fourth order linear problem. We extend this approach to pdes on evolving surfaces. We define an Eulerian level set method for partial differential equations on surfaces. The key idea is based on formulating the partial differential equation on all level set surfaces of a prescribed function $\\Phi$ whose zero level set is $\\Gamma$. We use Eulerian surface gradients to define weak forms

of elliptic operators which naturally generate weak formulations

of Eulerian elliptic and parabolic equations. This results in a degenerate equation formulated in anisotropic Sobolev spaces based on the level set function $\\Phi$. The resulting equation is then solved in one space dimension higher but can be solved on a fixed finite element grid.

Numerical experiments are described for several linear and Nonlinear partial differential equations. In particular the power of the method is demonstrated by employing it to solve highly nonlinear second and fourth order problems such as surface Allen-Cahn and Cahn-Hilliard equations and surface level set equations for geodesic mean curvature flow. In particular we show how surface level set and phase field models can be used to compute the motion of curves on surfaces. This is joint work with G. Dziuk(Freiburg).