论文信息 - Backward difference time discretization of parabolic differential equations on evolving surfaces

Backward difference time discretization of parabolic differential equations on evolving surfaces

A linear parabolic differential equation on a moving surface is discretized in space by evolving surface finite elements and in time by backward difference formulas (BDF). Using results from Dahlquist's G-stability theory and Nevanlinna & Odeh's multiplier technique together with properties of the spatial semi-discretization, stability of the full discretization is proven for the BDF methods up to order 5 and optimal-order convergence is shown. Numerical experiments illustrate the behaviour of the fully discrete method.

[1] Olavi Nevanlinna,et al. Multiplier techniques for linear multistep methods , 1981 .

[2] E. Hairer,et al. Solving Ordinary Differential Equations I , 1987 .

[3] Germund Dahlquist,et al. G-stability is equivalent toA-stability , 1978 .

[4] E. Hairer,et al. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[5] Charles M. Elliott,et al. Finite elements on evolving surfaces , 2007 .

[6] Charles M. Elliott,et al. L2-estimates for the evolving surface finite element method , 2012, Math. Comput..

[7] D. Gilbarg,et al. Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[8] C. M. Elliott,et al. A Fully Discrete Evolving Surface Finite Element Method , 2012, SIAM J. Numer. Anal..

[9] Gerhard Dziuk,et al. Runge–Kutta time discretization of parabolic differential equations on evolving surfaces , 2012 .

[10] M. Crouzeix,et al. On the equivalence of A-stability and G-stability , 1989 .