Analysis of Mixed Interior Penalty Discontinuous Galerkin Methods for the Cahn–Hilliard Equation and the Hele–Shaw Flow

Abstract

This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is based on fully implicit time-stepping and the other uses the energy-splitting timestepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on −1 only in some low polynomial orders, instead of exponential orders. Similar to [14], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space H 1 (Ω) and it is larger than the finite element space. This difficult is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved in [14]. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.the other hand, the majority of these works were done for a fixed parameter . The error bounds, which are obtained using the standard Gronwall inequality technique, show an exponential dependence on 1/ . Such an estimate is clearly not useful for small , in particular, in addressing the issue whether the computed numerical interfaces converge to the original sharp interface of the Hele-Shaw problem. Better and practical error bounds should only depend on 1/ in some (low) polynomial orders because they can be used to provide an answer to the above convergence question, which in fact is the best result (in terms of ) one can expect. The first such polynomial order in 1/ a priori estimate was obtained in [15] for mixed finite element approximations of the Cahn-Hilliard problem (1.1)-(1.5). In addition, polynomial order in 1/ a posteriori error estimates were obtained in [16] for the same mixed finite element methods. One of the key ideas employed in all these works is to use a nonstandard error estimate technique which is based on establishing a discrete spectrum estimate (using its continuous counterpart) for the linearized Cahn-Hilliard operator. An immediate corollary of the polynomial order in 1/ a priori and a posteriori error estimates is the convergence of the numerical interfaces of the underlying mixed finite element approximations to the Hele-Shaw flow before the onset of singularities of the Hele-Shaw flow as and mesh sizes h and k all tend to zero.The objectives of this paper are twofold: Firstly, we develop some MIP-DG methods and to establish polynomial order in 1/ a priori error bounds, as well as to prove convergence of numerical interfaces for the MIP-DG methods. This ...

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