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

TL;DR

This paper develops variational formulations for second-order elliptic PDEs in nondivergence form with Cordes coefficients, enabling finite element methods and adaptive algorithms with proven convergence.

Contribution

It introduces a new symmetric least-squares variational formulation for nondivergence form elliptic equations with Cordes coefficients, extending finite element analysis.

Findings

Established quasi-optimal a priori error bounds.

Proved convergence of an adaptive mesh-refinement algorithm.

Demonstrated numerical effectiveness on uniform and adaptive meshes.

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.