Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs
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For a well-posed non-selfadjoint indefinite second-order linear elliptic PDE with general coefficients 𝐀, 𝐛,γ in L^∞ and symmetric and uniformly positive definite coefficient matrix 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)× L^2 as well as in in L^2× L^2 up to oscillations. This generalises earlier contributions for piecewise Lipschitz continuous coefficients to L^∞ coefficients. The compactness argument of Schatz and Wang for the displacement-oriented problem does not apply immediately to the mixed formulation in H(div)× 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.
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