Abstract
We show that two widely used Galerkin formulations for second-order elliptic problems provide approximations which are actually superclose, that is, their difference converges faster than the corresponding errors. In the framework of linear elasticity, the two formulations correspond to using either the stiffness tensor or its inverse the compliance tensor. We find sufficient conditions, for a wide class of methods (including mixed and discontinuous Galerkin methods), which guarantee a supercloseness result. For example, for the HDG\(_{k}\) method using polynomial approximations of degree \({k>0}\), we find that the difference of approximate fluxes superconverges with order \({k+2}\) and that the difference of the scalar approximations superconverges with order \({k+3}\). We provide numerical results verifying our theoretical results.
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We wish to the thank one of the referees for constructive criticism leading to a better presentation of the material, including the discussion about the IP methods which we originally did not consider.
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Cockburn, B., Sánchez, M.A. & Xiong, C. Supercloseness of Primal-Dual Galerkin Approximations for Second Order Elliptic Problems. J Sci Comput 75, 376–394 (2018). https://doi.org/10.1007/s10915-017-0538-0
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DOI: https://doi.org/10.1007/s10915-017-0538-0