Inf-Sup Stability of Parabolic TraceFEM

TL;DR

This paper establishes a stable and well-posed TraceFEM method for the heat equation on surfaces, providing uniform stability and error estimates independent of surface cut configurations.

Contribution

It develops a parabolic inf-sup stability theory for TraceFEM with novel stabilizations, ensuring uniform stability and optimal error estimates on shape-regular meshes.

Findings

Proves necessary and sufficient conditions for inf-sup stability.

Shows uniform boundedness of key stability constants.

Derives optimal error estimates and regularity results.

Abstract

We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in $\mathbb{R}^n$. We consider the normal derivative volume stabilization and add an $L^2$-type stabilization to the time derivative. We assume that the representation of and the integration over the surface are exact, however, all our results are independent of how the surface cuts the bulk mesh. For any mesh for which the method is well-defined, we establish necessary and sufficient conditions for inf-sup stability of the proposed TraceFEM in terms of $H^1$-stability of a stabilized $L^2$-projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. We derive several consequences of uniform discrete inf-sup stability, namely uniform well-posedness, discrete maximal parabolic regularity, parabolic quasi-best approximation, convergence to minimal regularity solutions, and optimal order-regularity energy and $L^2 L^2$ error estimates. We show that the additional stabilization of the time derivative restores optimal conditioning of time-discrete TraceFEM typical of fitted discretizations.