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Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media

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Abstract

Asymptotic error expansions in the sense of L ∞-norm for the Raviart-Thomas mixed finite element approximation by the lowest-order rectangular element associated with a class of parabolic integro-differential equations on a rectangular domain are derived, such that the Richardson extrapolation of two different schemes and an interpolation defect correction can be applied to increase the accuracy of the approximations for both the vector field and the scalar field by the aid of an interpolation postprocessing technique, and the key point in deriving them is the establishment of the error estimates for the mixed regularized Green’s functions with memory terms presented in R. Ewing at al., Int. J. Numer. Anal. Model 2 (2005), 301–328. As a result of all these higher order numerical approximations, they can be used to generate a posteriori error estimators for this mixed finite element approximation.

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Authors and Affiliations

  1. School of Applied Mathematics, Central University of Finance and Economics, Beijing, 100081, P. R. China

    Shanghui Jia

  2. Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario, P7B 5E1, Canada

    Deli Li

  3. Department of Mathematics, Tianjin University of Finance and Economics, Tianjin, 300222, P. R. China

    Tang Liu & Shuhua Zhang

Authors
  1. Shanghui Jia
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  2. Deli Li
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  3. Tang Liu
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  4. Shuhua Zhang
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Correspondence to Shanghui Jia.

Additional information

This project was supported in part by the Special Funds for Major State Basic Research Project (2007CB8149), the National Natural Science Foundation of China (10471103 and 10771158), the Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), the NSERC, Tianjin Natural Science Foundation (07JCYBJC14300), and Tianjin University of Finance and Economics.

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Jia, S., Li, D., Liu, T. et al. Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media. Appl Math 53, 13–39 (2008). https://doi.org/10.1007/s10492-008-0011-3

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  • DOI: https://doi.org/10.1007/s10492-008-0011-3

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