Flux-mortar mixed finite element methods with multipoint flux approximation

TL;DR

This paper develops a flux-mortar mixed finite element method combined with multipoint flux approximation for Darcy flow, enabling efficient domain decomposition and interface coupling on non-matching grids, with proven well-posedness and error estimates.

Contribution

It introduces a novel flux-mortar mixed finite element approach with multipoint flux approximation for Darcy flow, including a new domain decomposition algorithm and preconditioner.

Findings

Method achieves accurate pressure and flux approximation.

Numerical experiments confirm robustness on complex porous media.

Efficient solver reduces computational cost.

Abstract

The flux-mortar mixed finite element method was recently developed for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating the method as a mixed finite element method with a quadrature rule. We develop a non-overlapping domain decomposition algorithm for the solution of the resulting algebraic system that reduces it to an interface problem for the flux-mortar, as well as an efficient interface preconditioner. A series of numerical experiments is presented illustrating the performance of the method on general grids, including applications to flow in complex porous media.