Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains
<|reference_start|>Robust Augmented Mixed Finite Element Methods for Stoke Interface Problems with Discontinuous Viscosity in Multiple Subdomains: 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.<|reference_end|>
@article{liang2024robust,
    title={Robust Augmented Mixed Finite Element Methods for Stoke Interface
  Problems with Discontinuous Viscosity in Multiple Subdomains},
    author={Yuxiang Liang and Shun Zhang},
    journal={arXiv preprint arXiv:2407.20655},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.20655},
    primaryClass={math.NA cs.NA}
}