Mathematics of Computation

Author(s): Malgorzata Peszynska.
Journal: Math. Comp. 65 (1996), 1019-1037.
MSC (1991): Primary 65M15; Secondary 45K05, 35K99, 76S05
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: Approximation of solutions to diffusion equations with memory represented by convolution integral terms is considered. Such problems arise from modeling of flows in fissured media. Convergence of the method is proved and results of numerical experiments confirming the theoretical results are presented. The advantages of implementation of the algorithm in a multiprocessing environment are discussed.

1.: T. Arbogast, Analysis of the Simulation of Single Phase Flow Through a Naturally Fractured Reservoir, SIAM J. Numer. Anal. 26 (1989), 12-29. MR 90e:76122
2.: T. Arbogast, J. Douglas (Jr.), Dual--Porosity Models for Flow in Naturally Fractured Reservoirs, In ``Dynamics of Fluids in Hierarchical Porous Media,'' J. H. Cushman, ed., Academic Press, London, 1990, 177--221.
3.: T. Arbogast, J. Douglas (Jr.), U. Hornung, Derivation of the Double Porosity Model of Single Phase Flow via Homogenization Theory, SIAM J. Math. Anal. 21 (1990), 823-836. MR 91d:76074
4.: Xiao Chuan Cai, Additive Schwarz algorithms for parabolic convection--diffusion equations, Numer. Math. 60 (1991), 41-61. MR 93a:65127
5.: Xiao Chuan Cai, W.D. Gropp, D.E. Keyes, Convergence rate estimate for a domain decomposition method, Numer. Math. 61 (1992), 153--169. MR 92k:65181
6.: C. Chen, V. Thomée, L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comp. 58 (1992), 587-602. MR 93g:65120
7.: P. G. Ciarlet, The Finite Element Method for Elliptic Problems, (1978) North--Holland. MR 58:25001
8.: M. Dryja, O. Widlund, An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Technical Report 339, Department of Computer Science, Courant Institute of Mathematical Sciences, New York, December 1987.
9.: M. Dryja, O.B. Widlund, Some Domain Decomposition Algorithms for Elliptic Problems, in: Iterative Methods for Large Linear Systems, Academic Press, 1990. CMP 90:07
10.: Domain Decomposition Methods for Partial Differential Equations, Proceedings of the First International Symposium on Domain Decomposition Methods for Partial Differential Equations, R. Glowinski, G. H. Golub, G. A. Meurant, J. Périaux eds., Paris, France, January, 1987, SIAM, Philadelphia, 1988.
11.: A. Greenbaum, Congming Li, Han Zheng Cao, Parallelizing Preconditioned Conjugate Gradient Algorithms, Technical Report, Courant Institute, 1988.
12.: G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990. MR 91c:45003
13.: K.H. Hoffmann, J. Zou, Parallel efficiency of domain decomposition methods, Parallel Computing 19 (1993), 1375-1392.
14.: U. Hornung, R. E. Showalter, Diffusion Models for Fractured Media, Jour. Math. Anal. Appl. 147 (1990), 69-80. MR 91d:76072
15.: Y. Lin, V. Thomée, L. B. Wahlbin, Ritz-Volterra projections to finite element spaces and applications to integrodifferential and related eqautions, SIAM J. Numer. Anal. 28 (1991), 1047-1070. MR 92k:65193
16.: P. Linz, Analytical and Numerical Methods for Volterra Equations, (1985) SIAM, Philadelphia. MR 86m:65163
17.: C. Lubich, Convolution Quadrature and Discretized Operational Calculus, Parts I & 2, Numer. Math. 52 (1988), 129-145 & 413-425. MR 89g:65018; MR 89g:65019
18.: R. C. MacCamy, J. S. Wong, Stability Theorems for Some Functional Equations, Trans. Amer. Math. Soc. 164 (1972), 1-37. MR 45:2432
19.: Maria-Luisa Mascarenhas, A linear homogenization problem with time dependent coefficient, Trans. Amer. Math. Soc 281 (1984), 179-195. MR 85c:45002
20.: V. McLean, V. Thomée, L.B. Wahlbin, Discretization with variable time steps of an evolution equation with a positive type memory term, Applied Mathematics Report AMRR 93.18, December 1993 School of Math., The University of New South Wales.
21.: R. K. Miller, An integrodifferential equation for rigid heat conductors with memory, Jour. Math. Anal. Appl. 66 (1978), 313-332. MR 80g:45015
22.: B. Neta, Numerical Solution of a Nonlinear Integro--differential Equation, Jour. Math. Anal. Appl. 89 (1982), 598-611. MR 84a:65105
23.: J. W. Nunziato, On heat conduction in materials with memory, Quarterly Appl. Math. 29 (1971), 187--204. MR 45:4749
24.: A.K. Pani, V. Thomée, L.B. Wahlbin, Numerical methods for hyperbolic and parabolic integrodifferential equations, J. Integral Equations Appl. 4 (1992), 533--584. MR 94c:65167
25.: M. Peszy\'{n}ska, Fluid Flow Through Fissured Media. Mathematical Analysis and Numerical Approach, Ph. D. Thesis (1992), University of Augsburg.
26.: M. Peszy\'{n}ska, Finite element approximation of a model of nonisothermal flow through fissured media, in: Finite Element Methods, M. K[?^?]ri[?^?]zek, P. Neittaanmäki, R. Stenberg (Eds), Marcel Dekker, 1994, 357--366.
27.: M. Peszy\'{n}ska, On a model for nonisothermal flow in fissured media, Differential Integral Equations 8 (1995), 1497--1516. CMP 95:12
28.: M. Peszy\'{n}ska, Analysis of an integro--differential equation arising from modelling of flows with fading memory through fissured media, J. Partial Diff. Eqs. 8 (1995), 159--173. MR 96a:45007
29.: A.H. Schatz, V. Thomée, W.L. Wendland, Mathematical Theory of Finite and Boundary Element Methods, Birkhäuser, Basel--Boston--Berlin, 1990. MR 92f:65004
30.: R. E. Showalter, Distributed Microstructure Models of Porous Media, in: ``Flow in Porous Media: proceedings of the Oberwolfach conference, June 21--27, 1992'', J. Douglas Jr. and U. Hornung, eds., Birkhäuser, Basel, 1993, 155--163. MR 95a:76091
31.: I. H. Sloan, V. Thomée, Time Discretization of an Integro--Differential Equation of Parabolic Type, SIAM J. Numer. Anal. 23 (1986) 1052--1061. MR 87j:65113
32.: L. Tartar, Nonlocal effects induced by homogenization, in: ``Partial Differential Equations and the Calculus of Variations, Essays in Honor of Ennio de Giorgi,'' F. Colombini et. al., eds., Birkhäuser, Boston, 1989, 925-938. MR 91c:35018
33.: L. Tartar, Memory effects and homogenization, Arch. Rat. Mech. Anal. 111 (1990), 121--133. MR 92h:35019
34.: V. Thomée, L. B. Wahlbin, Long time numerical solution of a parabolic equation with memory, Dept. of Math, Chalmers University of Technology, The University of Göteborg, Preprint No 1992-12/ISSN 0347-2809
35.: M. F. Wheeler, A Priori Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations, SIAM J. Numer. Anal., 10 (1973), 723-759. MR 50:3613
36.: Nai-ying Zhang, On fully discrete Galerkin approximations for partial integro--differential equations of parabolic type, Math. of Comp. 60 (1993) 133-166. MR 93d:65088

Retrieve articles in Mathematics of Computation with MSC (1991): 65M15, 45K05, 35K99, 76S05

Malgorzata Peszynska
Affiliation: Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6 01-447 Warszawa, Poland
Email: mpesz@ibspan.waw.pl

DOI: 10.1090/S0025-5718-96-00738-7
PII: S 0025-5718(96)00738-7
Keywords: Integro--partial differential equations, finite elements, convolution integrals
Received by editor(s): May 2, 1994
Received by editor(s) in revised form: August 2, 1994, October 25, 1994, February 12, 1995, and May 15, 1995
Copyright of article: Copyright 1996, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS