Robust augmented mixed FEMs for stokes interface problems with discontinuous viscosity in multiple subdomains

Abstract

A stationary Stokes problem with a piecewise constant viscosity coefficient with several 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 have the robust a priori error estimate is that two norms, one energy norm and one full norm, are used in the 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.