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Stability of mixed FEMs for non-selfadjoint indefinite second-order linear elliptic PDEs
Journal
Numerische Mathematik
ISSN
0029599X
Date Issued
2022-04-01
Author(s)
Carstensen, C.
Nataraj, Neela
Pani, Amiya K.
Abstract
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 and Brezzi-Douglas-Marini finite element families of any order and in any space dimension and leads to the best-approximation estimate in H(div)×L2 as well as in in L2× L2 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)×L2. But it allows the uniform approximation of some L2 contributions and can be combined with a recent L2 best-approximation result from the medius analysis. This technique circumvents any regularity assumption and the application of a Fortin interpolation operator.
Volume
150