WS, Ricardo Nochetto: TraceFEM: Ensuring and Circumventing Discrete Inf-Sup Conditions

Speaker
Ricardo Nochetto,University of Maryland at College Park

Abstract
For a $C^3$ stationary hypersurface embedded in $R^d$, which cuts an underlying shape-regular bulk mesh in an arbitrary manner, we present TraceFEMs with normal derivative volume stabilization for two distinct problems. We first develop a sharp parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation that adds a novel $L^2$-type stabilization to the time derivative. This yields uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, optimal order-regularity energy and $L^2 L^2$ error estimates, and robust conditioning with respect to time step. The second problem is a reformulation of the stationary surface Stokes equations as a non-symmetric indefinite elliptic problem governed by two Laplacians and further discretized by TraceFEM. We prove well-posedness, quasi-best approximation in a robust mesh-dependent $H^1$-norm for any polynomial degree, as well as optimal $L^2$ error estimates for both velocity and pressure. This entails a sufficiently small mesh size that solely depends on the Weingarten map and circumvents the usual discrete inf-sup condition. The first problem is joint with L. Bouck, V. Yushutin and M. Shakipov, while the second one is joint with M. Shakipov.