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Richardson extrapolation and defect correction of mixed finite element methods for integro-differential equations in porous media. (English) Zbl 1177.76408
Summary: Asymptotic error expansions in the sense of the $$L^{\infty }$$-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. The key point in deriving them is the establishment of the error estimates for mixed regularized Green functions with memory terms presented in [R. E. Ewing, Y. Lin, J. Wang and S. Zhang, Int. J. Numer. Anal. Model. 2, No. 3, 301–328 (2005; Zbl 1151.76594)]. 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.

##### MSC:
 76S05 Flows in porous media; filtration; seepage 45K05 Integro-partial differential equations 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Zbl 1151.76594
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