In this paper, both symmetric and nonsymmetric interior penalty discontinuous hpGalerkin 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 H 1 -norm, which are optimal in h and suboptimal in p, are derived. Moreover, on a regular mesh an hp-error estimate for the L 2 -norm is also established. Finally, numerical experiments illustrating the theoretical results are provided.
Introduction.In recent years, there has been renewed interest in discontinuous Galerkin methods for the numerical solution of a wide range of partial differential equations. This is due to their flexibility in local mesh adaptivity and their flexibility in handling nonuniform degrees of approximation for solutions whose smoothness exhibits variation over the computational domain. Based on Nitsche's symmetric formulation in 1970, these methods were introduced for second order elliptic and parabolic equations by Arnold [3], Douglas and Dupont [13], and Wheeler [24] and hence are presently called symmetric interior penalty discontinuous Galerkin (SIPG) methods. It is observed that SIPG methods are adjoint consistent, but the stabilizing parameters in these methods depend on the bounds of the coefficients of the problem considered and various constants involved in inverse inequalities. Recently, Oden, Babuška, and Baumann [21] proposed another discontinuous Galerkin method, which is based on a nonsymmetric formulation. Rivière, Wheeler, and Girault [23] and Houston, Schwab, and Süli [16] introduced and analyzed the nonsymmetric interior penalty discontinuous Galerkin (NIPG) method, which is a stabilized discontinuous Galerkin method. For a review, see [22], and for variants of discontinuous formulations, see Brezzi et al. [9], Arnold et al. [4], Houston, Robson, and Süli [17], and the references therein. A significant property of an NIPG method is that it is unconditionally stable with respect to the choice of the penalty parameter. Hence, this advantage has stimulated renewed interest in applying these methods to a large class of partial differential equations.In the literature, optimal a priori error estimates are derived in the broken H 1 -norm, and numerical experiments are conducted for SIPG and NIPG methods for linear self-adjoint elliptic problems; see [4], [23]. Except for [17], there are hardly any results on discontinuous Galerkin approximation of nonlinear elliptic problems. In [17], a one-parameter family of discontinuous Galerkin methods is applied to the
