Geometric PDE, Adaptive Methods, Finite Element Exterior Calculus, and Applications in Relativity
Thursday, December 15, 4PM – 5:15PM
We examine the theory and numerical treatment of coupled nonlinear elliptic geometric PDE containing critical exponents. A motivating example is the conformal formulation of the Einstein constraint equations. We first outline some new results on the existence of solutions to the constraint equations for rough metrics and arbitrarily prescribed mean extrinsic curvature. We then develop some new a priori error estimates for Galerkin finite element approximation; the estimates have the surprising feature that no angle conditions are involved in the case of critical and subcritical nonlinearity. Moreover, it then becomes possible to turn the a priori estimates into pointwise control of discrete solutions, without the need for a discrete maximum principle, and hence again without the need for angle conditions. We then briefly describe a new approach to analyzing the geometric error made if the domain is a smooth Riemannian manifold rather than a polyhedral domain. The approach involves the development of variational crimes analysis in Hilbert complexes, and then using the abstract framework to develop analogues of the Strang Lemmas for the Finite Element Exterior Calculus (FEEC). We indicate how this new variational crimes framework in FEEC recovers the classical a priori surface finite element estimates of Dziuk and Demlow, and allows for substantial generalizations, including hypersurfaces of arbitrary spatial dimension, the Hodge Laplacean, nonlinear problems, and evolution problems.
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