An adaptive octree finite element method for PDEs posed on surfaces

The paper develops a finite element method for partial differential equations posed on hypersurfaces in $\mathbb{R}^N$, $N=2,3$. The method uses traces of bulk finite element functions on a surface embedded in a volumetric domain. The bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures. The cartesian structure of the bulk mesh leads to easy and efficient adaptation process, while the trace finite element method makes fitting the mesh to the surface unnecessary. The number of degrees of freedom involved in computations is consistent with the two-dimension nature of surface PDEs. No parametrization of the surface is required; it can be given implicitly by a level set function. In practice, a variant of the marching cubes method is used to recover the surface with the second order accuracy. We prove the optimal order of accuracy for the trace finite element method in $H^1$ and $L^2$ surface norms for a problem with smooth solution and quasi-uniform mesh refinement. Experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom, if grid adaptation is based on an appropriate error indicator. The paper shows results of numerical experiments for a variety of geometries and problems, including advection-diffusion equations on surfaces. Analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted (trace) finite elements provide simple, efficient, and reliable tool for numerical treatment of PDEs posed on surfaces

Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem

This paper introduces bootstrap two-grid and multigrid finite element approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a closed surface. The proposed multigrid method is suitable for recovering eigenvalues having large multiplicity, computing interior eigenvalues, and approximating the shifted indefinite eigen-problem. Convergence analysis is carried out for a simplified two-grid algorithm and numerical experiments are presented to illustrate the basic components and ideas behind the overall bootstrap multigrid approach

Trace Finite Element Methods for PDEs on Surfaces

In this paper we consider a class of unfitted finite element methods for discretization of partial differential equations on surfaces. In this class of methods known as the Trace Finite Element Method (TraceFEM), restrictions or traces of background surface-independent finite element functions are used to approximate the solution of a PDE on a surface. We treat equations on steady and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in detail. We review the error analysis and algebraic properties of the method. The paper navigates through the known variants of the TraceFEM and the literature on the subject