Finite element approximation of diffusion equations with convolution terms
HTML articles powered by AMS MathViewer
- by Malgorzata Peszynska PDF
- Math. Comp. 65 (1996), 1019-1037 Request permission
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.References
- Todd Arbogast, Analysis of the simulation of single phase flow through a naturally fractured reservoir, SIAM J. Numer. Anal. 26 (1989), no. 1, 12–29. MR 977946, DOI 10.1137/0726002
- 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.
- Todd Arbogast, Jim Douglas Jr., and Ulrich Hornung, Derivation of the double porosity model of single phase flow via homogenization theory, SIAM J. Math. Anal. 21 (1990), no. 4, 823–836. MR 1052874, DOI 10.1137/0521046
- Xiao-Chuan Cai, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math. 60 (1991), no. 1, 41–61. MR 1131498, DOI 10.1007/BF01385713
- Xiao-Chuan Cai, William D. Gropp, and David E. Keyes, Convergence rate estimate for a domain decomposition method, Numer. Math. 61 (1992), no. 2, 153–169. MR 1147575, DOI 10.1007/BF01385503
- C. Chen, V. Thomée, and L. B. Wahlbin, Finite element approximation of a parabolic integro-differential equation with a weakly singular kernel, Math. Comp. 58 (1992), no. 198, 587–602. MR 1122059, DOI 10.1090/S0025-5718-1992-1122059-2
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- 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.
- M. Dryja, O.B. Widlund Some Domain Decomposition Algorithms for Elliptic Problems in: Iterative Methods for Large Linear Systems, Academic Press, 1990.
- 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.
- A. Greenbaum, Congming Li, Han Zheng Cao Parallelizing Preconditioned Conjugate Gradient Algorithms Technical Report, Courant Institute, 1988.
- G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
- K.H. Hoffmann, J. Zou Parallel efficiency of domain decomposition methods Parallel Computing 19 (1993), 1375-1392.
- Ulrich Hornung and Ralph E. Showalter, Diffusion models for fractured media, J. Math. Anal. Appl. 147 (1990), no. 1, 69–80. MR 1044687, DOI 10.1016/0022-247X(90)90385-S
- Yan Ping Lin, Vidar Thomée, and Lars B. Wahlbin, Ritz-Volterra projections to finite-element spaces and applications to integrodifferential and related equations, SIAM J. Numer. Anal. 28 (1991), no. 4, 1047–1070. MR 1111453, DOI 10.1137/0728056
- Peter Linz, Analytical and numerical methods for Volterra equations, SIAM Studies in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR 796318, DOI 10.1137/1.9781611970852
- C. Lubich, Convolution quadrature and discretized operational calculus. I, Numer. Math. 52 (1988), no. 2, 129–145. MR 923707, DOI 10.1007/BF01398686
- R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc. 164 (1972), 1–37. MR 293355, DOI 10.1090/S0002-9947-1972-0293355-X
- Maria Luisa Mascarenhas, A linear homogenization problem with time dependent coefficient, Trans. Amer. Math. Soc. 281 (1984), no. 1, 179–195. MR 719664, DOI 10.1090/S0002-9947-1984-0719664-3
- 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.
- R. K. Miller, An integro-differential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), no. 2, 313–332. MR 515894, DOI 10.1016/0022-247X(78)90234-2
- Beny Neta, Numerical solution of a nonlinear integro-differential equation, J. Math. Anal. Appl. 89 (1982), no. 2, 598–611. MR 677747, DOI 10.1016/0022-247X(82)90119-6
- Jace W. Nunziato, On heat conduction in materials with memory, Quart. Appl. Math. 29 (1971), 187–204. MR 295683, DOI 10.1090/S0033-569X-1971-0295683-6
- A. K. Pani, V. Thomée, and L. B. Wahlbin, Numerical methods for hyperbolic and parabolic integro-differential equations, J. Integral Equations Appl. 4 (1992), no. 4, 533–584. MR 1200801, DOI 10.1216/jiea/1181075713
- M. Peszyńska Fluid Flow Through Fissured Media. Mathematical Analysis and Numerical Approach Ph. D. Thesis (1992), University of Augsburg.
- M. Peszyńska Finite element approximation of a model of nonisothermal flow through fissured media in: Finite Element Methods, M. Kr̂iẑek, P. Neittaanmäki, R. Stenberg (Eds), Marcel Dekker, 1994, 357–366.
- M. Peszyńska On a model for nonisothermal flow in fissured media Differential Integral Equations 8 (1995), 1497–1516.
- Małgorzata Peszyńska, Analysis of an integro-differential equation arising from modelling of flows with fading memory through fissured media, J. Partial Differential Equations 8 (1995), no. 2, 159–173. MR 1331523
- Albert H. Schatz, Vidar Thomée, and Wolfgang L. Wendland, Mathematical theory of finite and boundary element methods, DMV Seminar, vol. 15, Birkhäuser Verlag, Basel, 1990. MR 1116555, DOI 10.1007/978-3-0348-7630-8
- Ralph E. Showalter, Distributed microstructure models of porous media, Flow in porous media (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 114, Birkhäuser, Basel, 1993, pp. 155–163. MR 1276501
- I. H. Sloan and V. Thomée, Time discretization of an integro-differential equation of parabolic type, SIAM J. Numer. Anal. 23 (1986), no. 5, 1052–1061. MR 859017, DOI 10.1137/0723073
- Luc Tartar, Nonlocal effects induced by homogenization, Partial differential equations and the calculus of variations, Vol. II, Progr. Nonlinear Differential Equations Appl., vol. 2, Birkhäuser Boston, Boston, MA, 1989, pp. 925–938. MR 1034036
- Luc Tartar, Memory effects and homogenization, Arch. Rational Mech. Anal. 111 (1990), no. 2, 121–133. MR 1057651, DOI 10.1007/BF00375404
- 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
- Mary Fanett Wheeler, A priori $L_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723–759. MR 351124, DOI 10.1137/0710062
- Nai Ying Zhang, On fully discrete Galerkin approximations for partial integro-differential equations of parabolic type, Math. Comp. 60 (1993), no. 201, 133–166. MR 1149295, DOI 10.1090/S0025-5718-1993-1149295-4
Additional Information
- Malgorzata Peszynska
- Affiliation: Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6 01-447 Warszawa, Poland
- Email: mpesz@ibspan.waw.pl
- 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 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 1019-1037
- MSC (1991): Primary 65M15; Secondary 45K05, 35K99, 76S05
- DOI: https://doi.org/10.1090/S0025-5718-96-00738-7
- MathSciNet review: 1344620