An interior penalty discontinuous Galerkin finite element method for quasilinear parabolic problems

•In this work we present an Interior Penalty Discontinuous Galerkin finite element method for solving quasilinear parabolic problems.•In these problems where the diffusion coefficient has a p-exponent form.•The natural formulation of the problem is in Lp space•For the IPDG method, stability bounds and a-priori error estimates are shown.•Two low computational cost Picard block-iterative methods are proposed.

In this paper, an Interior Penalty Discontinuous Galerkin (IPDG) finite element method is analyzed for approximating quasilinear parabolic equations. The equations can be characterized as perturbed parabolic p  -Laplacian equations. The fully discrete scheme is obtained by applying s-stage Diagonally Implicit Runge–Kutta (s-DIRK) methods for the time integration. The nonlinear systems of the algebraic equations appearing in s-DIRK cycles are solved by developing two low storage Picard iterative processes. A stability bound is shown for the semi-discrete IPDG solution in the broken ‖.‖DG,p-norm‖.‖DG,p-norm. Continuous in time a priori error estimates are proved in case of p>2p>2, when linear approximation space is used. A numerical test is performed in order to compare the performance of the two Picard iterative processes. Also, the results presented in the theoretical analysis are confirmed by numerical examples.