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

Abstract

AbstractFor a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients $${\mathbf {A}}, {\mathbf {b}},\gamma $$ A , b , γ in $$L^\infty $$ L ∞ and symmetric and uniformly positive definite coefficient matrix $${\mathbf {A}}$$ 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 and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in $$H({{\,\mathrm{div}\,}})\times L^2$$ H ( div ) × L 2 as well as in in $$L^2\times L^2$$ L 2 × L 2 up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to $$L^\infty $$ L ∞ coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in $$H({{\,\mathrm{div}\,}})\times L^2$$ H ( div ) × L 2 . But it allows the uniform approximation of some $$L^2$$ L 2 contributions and can be combined with a recent $$L^2$$ L 2 best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.