Convergence of a crystalline approximation for an area-preserving motion

• Published in: Journal of computational and applied mathematics Vol. 166; no. 2; pp. 427 - 452
• Main Authors: Ushijima, Takeo K., Yazaki, Shigetoshi
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15-04-2004; Elsevier
• Subjects: A priori estimate; Area-preserving; Convergence; Crystalline algorithm; Crystalline approximation; Crystalline curvature; Crystalline motion; Curve-shortening; Discrete version of Wirtinger's inequality; Discrete W1, p norm; Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Partial differential equations; Partial differential equations, boundary value problems; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Sciences and techniques of general use; Semi-discrete problem; Crystalline approximation; 53A04; 65M06; 45L05; 34A34; 34A99; 35R10; Discrete version of Wirtinger's inequality; Area-preserving; 65L; Convergence; Crystalline algorithm; A priori estimate; 41A25; Crystalline motion; Semi-discrete problem; Curve-shortening; Discrete W 1, p norm; 65D99; 45K05; Crystalline curvature; 65M12; Numerical approximation; Parabolic equation; Numerical analysis; Error estimation; Differential equation; Applied mathematics; Differential system; A priori estimation; Partial differential equation; Equation system
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2003.08.041

We consider an approximation of area-preserving motion in the plane by a generalized crystalline motion. The area-preserving motion is described by a parabolic partial differential equation with a nonlocal term, while the crystalline motion is governed by a system of ordinary differential equations. We show the convergence between these two motions. The convergence theorem is proved in two steps: first, an a priori estimate is established for a solution to the generalized crystalline motion; second, a discrete
W
1,
p
norms of the error is estimated for all 1⩽
p<∞ and, passing p to infinity, a discrete
W
1,∞ error estimate is obtained. We also construct an implicit scheme which enjoys several nice properties such as the area-preserving and curve-shortening, and compare our scheme with a simple scheme.