Flux-mortar mixed finite element methods on non-matching grids

TL;DR

This paper develops and analyzes a mortar mixed finite element method for Darcy flow on non-matching grids, introducing extension operators and demonstrating stability, convergence, and applicability to coupled Stokes-Darcy problems.

Contribution

It introduces a novel mortar technique with extension operators for mixed finite element approximation on non-matching grids, extending to saddle point problems.

Findings

The method is stable and convergent under certain conditions.

Extension operators effectively enforce weak pressure continuity.

Applicable to coupled Stokes-Darcy flow problems.

Abstract

We investigate a mortar technique for mixed finite element approximations of Darcy flow on non-matching grids in which the normal flux is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of pressure. In the mixed formulation of the problem, the normal flux is an essential boundary condition and it is incorporated with the use of suitable extension operators. Two such extension operators are considered and we analyze the resulting formulations with respect to stability and convergence. We further generalize the theoretical results, showing that the same domain decomposition technique is applicable to a class of saddle point problems satisfying mild assumptions. An example of coupled Stokes-Darcy flows is presented.