Efficient Solvers of Discontinuous Galerkin Discretization for the Cahn–Hilliard Equations

Abstract

In this paper, we develop and analyze a fast solver for the system of algebraic equations arising from the local discontinuous Galerkin (LDG) discretization and implicit time marching methods to the Cahn---Hilliard (CH) equations with constant and degenerate mobility. Explicit time marching methods for the CH equation will require severe time step restriction $$(\varDelta t \sim O(\varDelta x^4))$$ ( Δ t ~ O ( Δ x 4 ) ) , so implicit methods are used to remove time step restriction. Implicit methods will result in large system of algebraic equations and a fast solver is essential. The multigrid (MG) method is used to solve the algebraic equations efficiently. The Local Mode Analysis method is used to analyze the convergence behavior of the linear MG method. The discrete energy stability for the CH equations with a special homogeneous free energy density $$\Psi (u)=\frac{1}{4}(1-u^2)^2$$ ? ( u ) = 1 4 ( 1 ? u 2 ) 2 is proved based on the convex splitting method. We show that the number of iterations is independent of the problem size. Numerical results for one-dimensional, two-dimensional and three-dimensional cases are given to illustrate the efficiency of the methods. We numerically show the optimal complexity of the MG solver for $$\mathcal{P }^1$$ P 1 element. For $$\mathcal{P }^2$$ P 2 approximation, the optimal or sub-optimal complexity of the MG solver are numerically shown.