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**Superconvergence of mixed finite element approximations over quadrilaterals.**
*(English)*
Zbl 0926.65107

The objective of the paper is to investigate superconvergence phenomena for the approximate solution of second-order elliptic differential equations discretized by mixed finite element methods. The authors attention is focused on partitions of the space domain into convex quadrilaterals. The quadrilaterals elements are constructed by using local mapping techniques for any stable rectangular space such as Raviart-Thomas, Brezzi-Douglas-Fortin-Marini, etc.

A superconvergence result is established. The superconvergence indicates an accuracy of \(O(h^{k+2})\) for the mixed finite element approximation if the Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order \(k\) are employed with an optimal error estimate of \(O(h^{k+1})\). A part of the paper consists of a review of the construction of mixed elements over quadrilaterals as well as a general framework for the convergence and superconvergence analysis in mixed methods combined with some details. Numerical experiments are presented in order to illustrate the theoretical result.

The paper is interesting for numerical mathematicians dealing with mixed finite element methods and mainly for these of them dealing with mathematical modelling of fluid flow in porous media since the modelling process requires the determination of a very accurate fluid velocity.

A superconvergence result is established. The superconvergence indicates an accuracy of \(O(h^{k+2})\) for the mixed finite element approximation if the Raviart-Thomas or Brezzi-Douglas-Fortin-Marini elements of order \(k\) are employed with an optimal error estimate of \(O(h^{k+1})\). A part of the paper consists of a review of the construction of mixed elements over quadrilaterals as well as a general framework for the convergence and superconvergence analysis in mixed methods combined with some details. Numerical experiments are presented in order to illustrate the theoretical result.

The paper is interesting for numerical mathematicians dealing with mixed finite element methods and mainly for these of them dealing with mathematical modelling of fluid flow in porous media since the modelling process requires the determination of a very accurate fluid velocity.

Reviewer: K.Georgiev (Sofia)

### MSC:

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65N15 | Error bounds for boundary value problems involving PDEs |