Discontinuous Galerkin Methods for Quasi‐Linear Elliptic Problems of Nonmonotone Type

In this paper, both symmetric and nonsymmetric interior penalty discontinuous $hp$‐Galerkin methods are applied to a class of quasi‐linear elliptic problems which are of nonmonotone type. Using Brouwer’s fixed point theorem, it is shown that the discrete problem has a solution, and then, using Lipschitz continuity of the discrete solution map, uniqueness is also proved. A priori error estimates in the broken H¹‐norm, which are optimal in h and suboptimal in p, are derived. Moreover, on a regular mesh an $hp$‐error estimate for the L²‐norm is also established. Finally, numerical experiments illustrating the theoretical results are provided.