Finite element methods for second order linear elliptic partial differential equations in non-divergence form

Abstract:

This paper is concerned with finite element approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. A non-standard (primal) finite element method, which uses finite-dimensional subspaces consisting of globally continuous piecewise polynomial functions, is proposed and analyzed. The main novelty of the finite element method is to introduce an interior penalty term, which penalizes the jump of the flux across the interior element edges/faces, to augment a non-symmetric piecewise defined and PDE-induced bilinear form. Existence, uniqueness and error estimate in a discrete $W^{2,p}$ energy norm are proved for the proposed finite element method. This is achieved by establishing a discrete Calderon–Zygmund-type estimate and mimicking strong solution PDE techniques at the discrete level. Numerical experiments are provided to test the performance of proposed finite element methods and to validate the convergence theory.