Discontinuous Galerkin Finite Element Approximation of Nonlinear Second-Order Elliptic and Hyperbolic Systems

Abstract

We develop the convergence analysis of discontinuous Galerkin finite element approximations to symmetric second-order quasi-linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in $\mathbb{R}^d$, subject to mixed Dirichlet-Neumann boundary conditions. Optimal-order asymptotic bounds are derived on the discretization error in each case without requiring the global Lipschitz continuity or uniform monotonicity of the stress tensor. Instead, only local smoothness and a Garding inequality are used in the analysis.