On the error analysis of the time-continuous Discontinuous Galerkin scheme for degenerate parabolic equations
Abstract
We deal with an error analysis of a semidiscrete scheme for a doubly nonlinear parabolic partial differential equation (PDE), which admits the fast-diffusion type of degeneracy. The corresponding solutions of this equation are usually not smooth.
These equations have been studied in papers on the existence, uniqueness, and regularity of solutions to elliptic-parabolic differential equations (see, e.g., [1] and [2]). A well-known representative of such a PDE type is the Richards' equation, which is widely used to model porous media flow.
A vast number of numerical methods have been proposed for solving such problems. We mention [3], which demonstrated the higher-order space-time discontinuous Galerkin finite element method as a promising tool for solving of the Richards' equation in an efficient, robust and accurate way.
However, the rigorous mathematical theory for this method is still missing. Therefore, in this talk, we focus on the error analysis for the time-continuous scheme.
Due to its favorable features in this class of problems, we choose the incomplete interior penalty Galerkin (IIPG) method for spatial discretisation; cf. [3]. We pay special attention to the estimation of the accumulation term, which possibly can vanish.
Furthermore, we use continuous mathematical induction to derive a priori error estimates in the L2-norm and the so-called DG-norm with respect to the spatial discretisation parameter and the Hölder coefficient of the accumulation term derivative. [1] H.W. Alt, S.
Luckhaus, Quasilinear elliptic-parabolic differential equations. Math.
Z. 183 (1983), 311-342. [2] F. Otto , L1-contraction and uniqueness for quasilinear elliptic-parabolic equations.
J. Differ.
Equ. 131(1) (1996), 20-38. [3] V. Dolejší, M.
Kuraz, P. Solin, Adaptive higher-order space-time discontinuous Galerkin method for the computer simulation of variably-saturated porous media flows.
Appl. Math.
Model. (2019), 276-305