An edge-based pressure stabilisation technique for finite elements on arbitrarily anisotropic meshes

TL;DR

This paper introduces an edge-based pressure stabilization method for finite element analysis of the Stokes equations on highly anisotropic meshes, ensuring stability and convergence in complex moving domain applications.

Contribution

It develops a modified Continuous Interior Penalty stabilization technique that handles arbitrary anisotropies with proven stability and convergence properties.

Findings

Achieves ${ m O}(h^{3/2})$ convergence in energy norm

Achieves ${ m O}(h^{5/2})$ convergence in L2 norm

Numerical tests confirm theoretical stability and accuracy

Abstract

In this article, we analyse a stabilised equal-order finite element approximation for the Stokes equations on anisotropic meshes. In particular, we allow arbitrary anisotropies in a sub-domain, for example along the boundary of the domain, with the only condition that a maximum angle is fulfilled in each element.This discretisation is motivated by applications on moving domains as arising e.g. in fluid-structure interaction or multiphase-flow problems. To deal with the anisotropies, we define a modification of the original Continuous Interior Penalty stabilisation approach. We show analytically the discrete stability of the method and convergence of order ${\cal O}(h^{3/2})$ in the energy norm and ${\cal O}(h^{5/2})$ in the $L^2$-norm of the velocities. We present numerical examples for a linear Stokes problem and for a non-linear fluid-structure interaction problem, that substantiate the analytical results and show the capabilities of the approach.