Inf-sup stability of the trace 𝐏₂–𝐏₁ Taylor–Hood elements for surface PDEs

Olshanskii, Maxim; Reusken, Arnold; Zhiliakov, Alexander

doi:10.1090/mcom/3551

Inf-sup stability of the trace $\mathbf {P}_2$–$P_1$ Taylor–Hood elements for surface PDEs
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by Maxim A. Olshanskii, Arnold Reusken and Alexander Zhiliakov;

Math. Comp. 90 (2021), 1527-1555

DOI: https://doi.org/10.1090/mcom/3551

Published electronically: March 16, 2021

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Abstract:

The paper studies a geometrically unfitted finite element method (FEM), known as trace FEM or cut FEM, for the numerical solution of the Stokes system posed on a closed smooth surface. A trace FEM based on standard Taylor–Hood (continuous $\mathbf P_2$–$P_1$) bulk elements is proposed. A so-called volume normal derivative stabilization, known from the literature on trace FEM, is an essential ingredient of this method. The key result proved in the paper is an inf-sup stability of the trace $\mathbf P_2$–$P_1$ finite element pair, with the stability constant uniformly bounded with respect to the discretization parameter and the position of the surface in the bulk mesh. Optimal order convergence of a consistent variant of the FEM follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples.

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