ADFE METHOD WITH HIGH ACCURACY FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

Acta mathematica scientia,Series B ›› 2002, Vol. 22 ›› Issue (4): 473-483.

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ADFE METHOD WITH HIGH ACCURACY FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

CUI Xia

Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics,
P.O. Box 8009-26, Beijing 100088, China

Online:2002-10-14 Published:2002-10-14
Supported by:
Supported by China National Key Program for Developing Basic Sciences (G1999032801), Mathematical Tianyuan Foundation and NNSF of China (19932010).

Abstract

Abstract:

Alternating direction finite element (ADFE) scheme for d-dimensional nonlin-ear system of parabolic integro-differential equations is studied. By using a local approxi-mation based on patches of finite elements to treat the capacity term qi(u), decomposition of the coefficient matrix is realized; by using alternating direction, the multi-dimensional problem is reduced to a family of single space variable problems, calculation work is sim-
plified; by using finite element method, high accuracy for space variant is kept; by using inductive hypothesis reasoning, the difficulty coming from the nonlinearity of the coeffi-cients and boundary conditions is treated; by introducing Ritz-Volterra projection, the difficulty coming from the memory term is solved. Finally, by using various techniques for priori estimate for differential equations, the unique resolvability and convergence proper-
ties for both FE and ADFE schemes are rigorously demonstrated, and optimal H1 and L2 norm space estimates and O((t)2) estimate for time variant are obtained.

Key words: Parabolic integro-differential system, nonlinear, alternating direction, finite element, high accuracy

CLC Number:

65M60

CUI Xia. ADFE METHOD WITH HIGH ACCURACY FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS[J].Acta mathematica scientia,Series B, 2002, 22(4): 473-483.

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ADFE METHOD WITH HIGH ACCURACY FOR NONLINEAR PARABOLIC INTEGRO-DIFFERENTIAL SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

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