A Stabilized Finite Element Method for the Darcy Problem on Surfaces

TL;DR

This paper introduces a stabilized finite element method for solving the Darcy problem on surfaces, utilizing weak enforcement of tangential velocity conditions and standard polynomial spaces, with proven optimal error estimates.

Contribution

It presents a novel stabilized finite element approach for surface Darcy problems that simplifies implementation and provides rigorous error analysis.

Findings

Achieves optimal a priori error estimates

Enforces tangential velocity weakly via bilinear form

Uses standard polynomial spaces on triangulations

Abstract

We consider a stabilized finite element method for the Darcy problem on a surface based on the Masud-Hughes formulation. A special feature of the method is that the tangential condition of the velocity field is weakly enforced through the bilinear form and that standard parametric continuous polynomial spaces on triangulations can be used. We prove optimal order a priori estimates that take the approximation of the geometry and the solution into account.