Discrete Miranda-Talenti estimates and applications to linear and nonlinear PDEs

In this article, we construct simple and convergent finite element methods for linear and nonlinear elliptic differential equations in non-divergence form with discontinuous coefficients. The methods are motivated by discrete Miranda-Talenti estimates, which relate the H-2 semi-norm of piecewise polynomials with the L-2 norm of its Laplacian on convex domains. We develop a stability and convergence theory of the proposed methods, and back up the theory with numerical experiments. (C) 2019 Elsevier B.V. All rights reserved.