Finite element methods for surface PDEs

In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples

The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations

The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad class of finite difference methods combined with cutoff for linear parabolic equations. Two applications are investigated, linear anisotropic diffusion problems satisfying the setting of the convergence analysis and nonlinear lubrication-type equations for which it is unclear if the convergence analysis applies. The numerical results are shown to be consistent with the theory and in good agreement with existing results in the literature. The convergence analysis and applications demonstrate that the cutoff method is an effective tool for use in the computation of nonnegative solutions. Cutoff can also be used with other discretization methods such as collocation, finite volume, finite element, and spectral methods and for the computation of positive solutions.Comment: 19 pages, 41 figure

Finite element approximation of a sixth order nonlinear degenerate parabolic equation

We consider a finite element approximation of the sixth order nonlinear degenerate parabolic equation ut = ?.( b(u)? 2u), where generically b(u) := |u|? for any given ? ? (0,?). In addition to showing well-posedness of our approximation, we prove convergence in space dimensions d ? 3. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented