A priori error analysis of a Local Discontinuous Galerkin time-continuous scheme for a nonlinear degenerate parabolic equation modeling porous media flows
Abstract
We study an error analysis of a semidiscrete scheme for a doubly nonlinear parabolic partial differential equation, which can admit the fast-diffusion type of degeneracy. Consequently, the solution to this problem is not regular. Moreover, its existence, uniqueness, and regularity have been studied in [1, 4]. A typical example of this class of problems is Richards' equation, widely used in the modeling of flows in porous media. Many numerical methods have been suggested to solve such problems. In [5], the higher-order space-time discontinuous Galerkin finite element method has been applied and shown great performance in the sense of efficiency, accuracy, and robustness. However, the corresponding rigorous mathematical theory has not been provided yet. This talk presents the a priori error analysis for the time-continuous scheme. Due to the presence of nonlinearities for the spatial discretization, we choose the Local Discontinuous Galerkin method [2]. Thus, instead of the original problem, we consider the expanded mixed formulation [3] and define the method. Moreover, since the considered problem has an additional nonlinearity, special techniques are applied to derive the a priori error bounds. In particular, we give error estimates in L2-norm and the jump form with respect to the spatial discretization parameter and the Holder coefficient of the nonlinear term derivative. Numerical examples accompany the proposed theory.
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