Andrew Gillette, University of California, San Diego

In this talk, I will discuss two trends in the analysis of finite element methods (FEMs) for modern applications: (1) the use of polygonal or polyhedral mesh elements for domain decomposition and (2) the use of reduced ``serendipity'' basis sets for efficient solution approximation. I will show how the error analysis of FEMs employing generalized barycentric coordinates (GBCs) on meshes of polygons or polyhedra requires a subtle combination of computational geometry and functional analysis.  The derivation of a linear order error estimate will be shown to be subject to certain geometric constraints, depending on the type of GBCs being used. Additionally, I will explain the construction of ``serendipity'' FEMs in which higher order convergence rates can be obtained with fewer than the traditionally expected number of degrees of freedom. These techniques have enabled a joint project with the National Biomedical Computation Resource at UC San Diego to accelerate cardiac electrophysiology simulations; initial results will be presented.