Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains

TL;DR

This paper introduces a new augmented mixed finite element method for Stokes interface problems with discontinuous viscosity across multiple subdomains, ensuring robustness without the need for a robust inf-sup condition.

Contribution

It proposes an ultra-weak augmented mixed formulation combined with a Galerkin-least-squares approach to achieve stability and robustness in complex multi-subdomain Stokes problems.

Findings

Achieves robust a priori error estimates in energy norm

Validates the method with extensive numerical tests

Introduces a singular Kellogg-type exact solution example

Abstract

A stationary Stokes problem with a piecewise constant viscosity coefficient in multiple subdomains is considered in the paper. For standard finite element pairs, a robust inf-sup condition is required to show the robustness of the discretization error with respect to the discontinuous viscosity, which has only been proven for the two-subdomain case in the paper [Numer. Math. (2006) 103: 129--149]. To avoid the robust inf-sup condition of a discrete finite element pair for multiple subdomains, we propose an ultra-weak augmented mixed finite element formulation. By adopting a Galerkin-least-squares method, the augmented mixed formulation can achieve stability without relying on the inf-sup condition in both continuous and discrete settings. The key step to having a robust priori error estimate is to use two norms, one energy norm and one full norm, in robust continuity. The robust coercivity is proved for the energy norm. A robust a priori error estimate in the energy norm is then derived with the best approximation property in the full norm for the case of multiple subdomains. Additionally, the paper introduces a singular Kellogg-type example with exact solutions for the first time. Extensive numerical tests are conducted to validate the robust error estimate.