Discrete Miranda-Talenti estimates and applications to linear and nonlinear PDEs (Q2423689)

The paper is concerned with the numerical solution of second-order elliptic equations in non-divergent form with non-differentiable coefficients. In this case, the variational formulation cannot be derived by the standard approach using Green's theorem. The authors focus on the case where a domain \(\Omega\) is convex and the coefficient matrix is possibly discontinuous, but essentially bounded and satisfies the Cordes condition. It is known that then there exists a strong solution \(u \in H^2(\Omega)\). The authors derive the discrete Miranda-Talenti inequality, which relates the \(L^2\)-norm and the \(H^2\)-seminorm for a function \(v_h\) from the Lagrange finite element space. Based on this inequality, they propose the finite element method for linear elliptic problems in non-divergence form and derive optimal error estimates. The results are then extended to the nonlinear Hamilton-Jacobi-Bellman equation. Numerical experiments are provided to confirm the theoretical results and to illustrate the efficiency and applicability of the method.