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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

##### MSC:
 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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