492 | IGPM492.pdf September 2019 |

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

AUTHORS | Maxim A. Olshanskii, Arnold Reusken, Alexander Zhiliakov |

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 P2-P1) 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 P2-P1 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 finite element method follows from this new stability result and interpolation properties of the trace FEM. Properties of the method are illustrated with numerical examples. |

KEYWORDS | surface Stokes problem, trace finite element method, Taylor–Hood elements, material surfaces, fluidic membranes |

DOI | 10.1090/mcom/3551 |

PUBLICATION | Mathematics of Computation 90 (2021), 1527-1555 |