Superconvergence of H¹-Galerkin mixed finite element methods for parabolic problems

Abstract

In this article, we study the semidiscrete H ¹-Galerkin mixed finite element method for parabolic problems over rectangular partitions. The well-known optimal order error estimate in the L ²-norm for the flux is of order 𝒪(h ^k+1) (SIAM J. Numer. Anal. 35 (2), (1998), pp. 712–727), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas element. We derive a superconvergence estimate of order 𝒪(h ^k+3) between the H ¹-Galerkin mixed finite element approximation and an appropriately defined local projection of the flux variable when k ≥ 1. A the new approximate solution for the flux with superconvergence of order 𝒪(h ^k+3) is realized via a postprocessing technique using local projection methods.

Keywords:

• parabolic problem
• H ¹-Galerkin mixed finite element method
• superconvergence
• error estimates

AMS Subject Classifications::

• 65M60
• 65M15
• 65M12

Acknowledgement

The authors wish to thank the referee for giving valuable suggestions to the manuscript.