A nonconforming saddle point least squares approach for elliptic interface problems

TL;DR

This paper introduces a non-conforming saddle point least squares method for second order elliptic interface problems with discontinuous coefficients, enabling higher order flux approximation and efficient iterative solvers.

Contribution

It extends the SPLS framework to non-conforming finite element trial spaces with local projections, ensuring automatic stability and improved flux approximation.

Findings

Automatic satisfaction of the inf-sup condition.

Effective local projection-based trial spaces.

Numerical validation in 2D and 3D domains.

Abstract

We present a non-conforming least squares method for approximating solutions of second order elliptic problems with discontinuous coefficients. The method is based on a general Saddle Point Least Squares (SPLS) method introduced in previous work based on conforming discrete spaces. The SPLS method has the advantage that a discrete $\inf-\sup$ condition is automatically satisfied for standard choices of test and trial spaces. We explore the SPLS method for non-conforming finite element trial spaces which allow higher order approximation of the fluxes. For the proposed iterative solvers, inversion at each step requires bases only for the test spaces. We focus on using projection trial spaces with local projections that are easy to compute. The choice of the local projections for the trial space can be combined with classical gradient recovery techniques to lead to quasi-optimal approximations of the global flux. Numerical results for 2D and 3D domains are included to support the proposed method.