Discontinuous Galerkin method for nonlinear parabolic problems with mixed boundary condition (Q2927960)

The present authors [J. Math. Anal. Appl. 315, No. 1, 132--143 (2006; Zbl 1094.65091)] applied the discontinuous Galerkin (DG) method to parabolic problems with homogeneous Neumann boundary condition and constructed DG spatial discretized approximation and obtained an optimal \(L^\infty (L^2)\) error estimate. In another paper [J. Appl. Math. Inform. 28, No. 3--4, 953--966 (2010; Zbl 1294.65092)], they applied the DG method to construct the fully discrete approximations for parabolic problems with homogeneous Neumann boundary condition and obtain an optimal order of convergence in the \(l^\infty (L^2)\) normed space. In this paper the authors require very weak conditions on the terms characterizing the nonlinearity of the parabolic problem. It is also observed that the parabolic problem studied in this paper is related with mixed nonhomogeneous Dirichlet-nonhomogeneous Neumann boundary conditions.