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Finite element approximation of diffusion equations with convolution terms

Author: Malgorzata Peszynska
Journal: Math. Comp. 65 (1996), 1019-1037
MSC (1991): Primary 65M15; Secondary 45K05, 35K99, 76S05
MathSciNet review: 1344620
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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.

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  • 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 $L_2$ 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

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Additional Information

Malgorzata Peszynska
Affiliation: Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6 01-447 Warszawa, Poland

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
Article copyright: © Copyright 1996 American Mathematical Society

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