Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs

TL;DR

This paper proves the stability and best-approximation properties of mixed finite element methods for non-selfadjoint indefinite elliptic PDEs with $L^ abla$ coefficients, extending previous results to more general coefficient classes.

Contribution

It establishes the unique solvability and uniform boundedness of mixed FEMs for a broad class of indefinite elliptic PDEs with $L^ abla$ coefficients, generalizing earlier work.

Findings

Mixed FEMs are uniquely solvable and uniformly bounded for sufficiently fine meshes.

The results apply to Raviart-Thomas and Brezzi-Douglas-Marini elements of any order and dimension.

The paper achieves optimal approximation estimates without regularity assumptions.

Abstract

For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients $\mathbf A, \mathbf b,\gamma$ in $L^\infty$ and symmetric and uniformly positive definite coefficient matrix $\mathbf A$, this paper proves that mixed finite element problems are uniquely solvable and the discrete solutions are uniformly bounded, whenever the underlying shape-regular triangulation is sufficiently fine. This applies to the Raviart-Thomas (RT) and Brezzi-Douglas-Marini (BDM) finite element families of any order and in any space dimension and leads to the best-approximation estimate in $H(div)\times L^2$ as well as in in $L^2\times L^2$ up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to $L^\infty$ coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in $H(div)\times L^2$. But it allows the uniform approximation of some $L^2$ contributions and can be combined with a recent $L^2$ best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.