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Abstract
It is known that the zero level set of the solution of the Allen-Cahn equation ut - △u =ε-2(u - u3) approaches, as ε→ 0, to the solution of the motion by mean curvature flow. In this paper, we consider the convergence of numerical solutions of a discretized Allen-Cahn equation. In particular,we show that if for some p > 1, the space mesh size △x satisfies △z = O(εp), and the time mesh size satisfies the Courant-Friedrich.
1Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA. The author thanks the financial support of the Alfred P. Sloan Research Fellowship and the National Science Foundation Grant DMS-9404773
2Center for Mathematics Analysis and It's Applications, School of Mathematical Sciences, University of Sussex, Brighton, BN1 SQH, UK
2Center for Mathematics Analysis and It's Applications, School of Mathematical Sciences, University of Sussex, Brighton, BN1 SQH, UK
3Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI 48128, USA.
1Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA. The author thanks the financial support of the Alfred P. Sloan Research Fellowship and the National Science Foundation Grant DMS-9404773
2Center for Mathematics Analysis and It's Applications, School of Mathematical Sciences, University of Sussex, Brighton, BN1 SQH, UK
2Center for Mathematics Analysis and It's Applications, School of Mathematical Sciences, University of Sussex, Brighton, BN1 SQH, UK
3Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI 48128, USA.
Notes
1Department of Mathematics and Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA. The author thanks the financial support of the Alfred P. Sloan Research Fellowship and the National Science Foundation Grant DMS-9404773
2Center for Mathematics Analysis and It's Applications, School of Mathematical Sciences, University of Sussex, Brighton, BN1 SQH, UK
2Center for Mathematics Analysis and It's Applications, School of Mathematical Sciences, University of Sussex, Brighton, BN1 SQH, UK
3Department of Mathematics and Statistics, University of Michigan-Dearborn, Dearborn, MI 48128, USA.