Error Estimates with Smooth and Nonsmooth Data for a Finite Element Method for the Cahn-Hilliard Equation

A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidisCrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate. The Cahn-Hilliard equation where typically @(u) = u3 u , together with appropriate boundary and initial conditions, is a phenomenological model for phase separation and spinodal decomposition. The boundary conditions are such that the fourth-order differential operator in (1.1) can be written as the square of a second-order elliptic operator. Relying on this fact, we study numerical schemes for (1.1), which for the approximation of the spatial variables are based on standard Galerkin finite element methods for second-order elliptic problems. We discuss spatially semidiscrete schemes as well as a completely discrete scheme based on the backward Euler method. A semidiscrete finite element method (with numerical quadrature) of this type for the Cahn-Hilliard equation was first introduced and analyzed by Elliott, French, and Milner [7]. Completely discrete schemes based on the same idea Received March 19, 199 1. 1991 Mathematics Subject Classijication. Primary 65M 15, 65M60.