A nonconforming finite-element method for the two-dimensional Cahn- Hilliard equation. (English) Zbl 0686.65086

The paper deals with the equation \((1)\quad u_ t+\Delta^ 2u=\Delta \Phi (u)\) in \(\Omega \times (0,T)\) subject to the boundary conditions \((2)\quad \partial u/\partial \nu |_{\partial \Omega}=0,\quad \partial /\partial \nu (\Phi (u)-\Delta u)|_{\partial \Omega}=0\) and the initial conditions \((3)\quad u(.,0)=u_ 0.\) Here \(\Omega\) is a rectangle and \(\nu\) denotes the outward pointing normal to \(\partial \Omega\). A semidiscrete (with continuous time) Galerkin approximation of (1)-(3) is considered using the finite-element space consisting of Morley’s nonconforming shape functions \(\chi \in L^{\infty}(\Omega)\) defined as follows: given a triangulation \(T_ h\) consisting of triangles, then \(\chi\) is continuous at the vertices of \(T_ h\), the normal derivative \(\partial \chi /\partial \nu\) is continuous at the midpoints of all edges of the triangles \(\tau \in T_ h\) and \(\chi |_{\tau}\) is a quadratic polynomial for each \(\tau \in T_ h\). The estimate of the error is proved assuming some regularity of functions \(u_ 0\) and \(\Phi\).
Reviewer: H.Marcinkowska


65N40 Method of lines for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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