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arXiv · 1606.05631

Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients

Dietmar Gallistl
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Abstract

This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\ 51(2013), pp.\ 2088--2106.], and the second one is a new symmetric formulation based on a least-squares functional. These formulations enable the use of standard finite element techniques for variational problems in subspaces of $H^2$ as well as mixed finite element methods from the context of fluid computations. Besides the immediate quasi-optimal a~priori error bounds, the variational setting allows for a~posteriori error control with explicit constants and adaptive mesh-refinement. The convergence of an adaptive algorithm is proved. Numerical results on uniform and adaptive meshes are included.

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Dietmar Gallistl. 2016-06-17. Variational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients. https://arxiv.org/abs/1606.05631

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