A unified analysis of the local discontinuous Galerkin method for a class of nonlinear problems

In this paper we analyze the main features of the local discontinuous Galerkin method applied to nonlinear boundary value problems in the plane. We consider a class of nonlinear elliptic problems arising in heat conduction and fluid mechanics. The approach, which has been originally applied to several linear boundary value problems, is based on the introduction of additional unknowns given by the flux and the gradient of the temperature (velocity) for diffusion problems (fluid mechanics), and considers convex and nonconvex bounded domains with polygonal boundaries. Our present analysis unifies and simplifies the derivation of the results given in previous works. Several numerical examples are presented, which validate our theoretical results.