CVC Seminar: Wed Mar 23 1:30-3pm ACES 4.304

Dr. Michael Holst

Convergence of Adaptive Finite Element Methods for Nonlinear Geometric PDE

There is currently tremendous interest in geometric PDE, due in part to activities such as the geometric flow program used recently to solve the Poincare conjecture, and the NSF-funded LIGO project for the detection of gravitational wave emission and propagation involving numerical solution of the Einstein equations. We motivate things by examining the particular geometric elliptic system forming the constraints in the Einstein equations, and summarize some basic results about solutions. We then develop a weak-* convergence framework for abstract approximation algorithms for nonlinear operator equations. We next introduce some additional conditions that allow for building a contraction framework. We then indicate how both convergence frameworks can be used to recover a number of existing convergence and optimality results for adaptive finite element methods for linear and nonlinear elliptic problems on open sets in Rn, as well as establish some new results for geometric elliptic PDE problems on Riemannian manifolds. Time-permitting, we outline a new analysis of surface finite element methods for problems such as the Einstein constraints, by first analyzing variational crimes in Hilbert complexes, and then applying the results to finite element exterior calculus on hypersurfaces.

Parts of this work are joint work with Gabriel Nagy, Gantumur Tsogtgerel, Yunrong Zhu, Ari Stern, and Ryan Szypowski [1-4].

[1] M. Holst, G. Tsogtgerel, Y. Zhu, “Local Convergence of Adaptive Methods for Nonlinear Partial Differential Equations”, Submitted for publication. http://arxiv.org/abs/1001.1382

[2] M. Holst and A. Stern, “Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces”, Submitted for publication. http://arxiv.org/abs/1005.4455

[3] M. Holst and A. Stern, “Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation”, Submitted for publication. http://arxiv.org/abs/1010.6127

[4] M. Holst, R. Szypowski, G. Tsogtgerel, and Y. Zhu, “Convergence Adaptive Finite Element Approximation of the Einstein Constraints”, Preprint.