Partial differential equations on surfaces
Charlie Elliott (University of Sussex, UK)
In these notes we consider elliptic and parabolic partial differential equations on hypersurfaces G e R^{n+1}. Two approaches will be discussed. In the first we use surface gradients to define weak forms of elliptic operators and naturally generate weak formulations of elliptic and parabolic equations on G. In the second approach we define an Eulerian level set method for partial differential equations on a hypersurface G contained in a domain W e R^{n+1}. The key idea is based on formulating the partial differential equations on all level set surfaces of a prescribed function F whose zero level set is G. The resulting equation involves degenerate elliptic operator but can be solved in W which although one dimension higher does not explicitly refer to the hypersurface $\Gamma$ .
In both approaches we formulate a scalar conservation law on stationary and evolving hypersurfaces G(t) and, in the case of a diffusive flux, derive a surface transport and diffusion equation. Our motivation is to formulate finite element approximations. In the first approach the key idea is based on the approximation of G by a polyhedral surface G_h consisting of a union of simplices (triangles for n=2, intervals for n=1) with vertices on G. A finite element space of functions is then defined by taking the continuous functions on G_h which are linear affine on each simplex of the polygonal surface. Our surface finite element method (SFEM) or evolving surface finite element method (ESFEM) is applied to weak forms of the equations. The computation of the mass and element stiffness matrices are simple and straightforward. In the second approach finite element method is applied to the weak form of the conservation equation yielding an Eulerian FEM. The computation of the mass and element stiffness matrices are simple and straightforward.
Numerical experiments are described for several linear and nonlinear partial differential equations. We describe how this framework may be employed in applications. 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.
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