Error estimates and numerical simulations of mean curvature flow through a modified reaction-diffusion equation

• Published in: Numerical functional analysis and optimization Vol. 19; no. 5-6; pp. 513 - 528
• Main Authors: Fierro, F., Goglione, R.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Marcel Dekker, Inc 01.01.1998; Taylor & Francis
• Subjects: 1991 Mathematics Subject Classification 35A35; 1991 Mathematics Subject Classification 35B25; 1991 Mathematics Subject Classification 35K57; 1991 Mathematics Subject Classification 49Q05; 1991 Mathematics Subject Classification 65M60; Exact sciences and technology; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical simulation; Partial differential equations, miscellaneous problems; Sciences and techniques of general use; Flow simulation; Error estimation; Singular perturbation; Interaction potential; Numerical simulation; Approximation theory; Reaction diffusion equation; Curvature; Partial differential equation; Interface
• ISSN: 0163-0563; 1532-2467
• DOI: 10.1080/01630569808816842

We consider the following singularly perturbed reaction-diffusion equation with a small parameter € > 0 and double obstacle potential ψ. The zero level set of approximates the evolution of a surface of codimension 1, which moves in its inner normal direction with velocity This approach is insensitive to the choice of the potential ψ and thus differs from the usual Allen-Cahn approximation. We prove optimal interface error estimates of order O(€ 2 ), for smooth flows. To this aim we construct proper barriers dictated by formal asymptotics, we introduce a modified distance function, and we exploit the validity of a comparison lemma. Finally we present some numerical results.