The state-of-the art proof of a global inf-sup condition on mixed finite element schemes does not allow for an analysis of truly indefinite, second-order linear elliptic PDEs. This paper, therefore, first analyses a nonconforming finite element discretization which converges owing to some a priori \(L^2\) error estimates even for reduced regularity on non-convex polygonal domains. An equivalence result of that nonconforming finite element scheme to the mixed finite element method (MFEM) leads to the well-posedness of the discrete solution and to a priori error estimates for the MFEM. The explicit residual-based a posteriori error analysis allows some reliable and efficient error control and motivates some adaptive discretization which improves the empirical convergence rates in three computational benchmarks.
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The first author acknowledges the support of National Program on Differential Equations: Theory, Computation & Applications (NPDE-TCA) vide Department of Science & Technology (DST) Project No. SR/S4/MS:639/09 during his visit to IIT Bombay. The second author acknowledges the financial support of Council of Scientific and Industrial Research (CSIR), Government of India. The authors sincerely thank the two anonymous referees for careful reviews and several remarks which improved the paper.
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Carstensen, C., Dond, A.K., Nataraj, N. et al. Error analysis of nonconforming and mixed FEMs for second-order linear non-selfadjoint and indefinite elliptic problems. Numer. Math. 133, 557–597 (2016). https://doi.org/10.1007/s00211-015-0755-0
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