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Adaptive Multilevel Finite Element Methods on Manifolds with Applications in Relativity Physics and Biology

Michael Holst

Presented at the 1997 CRPC Annual Meeting Poster Session

In this talk, we consider the numerical treatment of coupled nonlinear elliptic systems on manifolds, which arise for example in general relativity and in models of biological membranes. We begin by taking a brief look at these applications, and by reviewing some concepts from differential geometry. Weak formulations of covariant nonlinear elliptic systems on compact manifolds are then examined. Finite element approximation theory on manifolds is discussed, and a computer implementation MC (manifold code) is described. MC is a simplex-based finite element C++ code for the numerical treatment of covariant differential operators on 2- and 3-manifolds. In addition to this differential geometric framework, MC implements a robust posteriori error estimation, adaptive conforming subdivision of simplices, decoupling methods, global inexact-Newton methods, continuation methods, and multilevel methods. We describe some of these algorithms in MC and some of its other features, and present some initial numerical experiments for a biological membrane problem, and for the elliptic constraints in the Einstein equations in a general setting.