Extrapolation for the Second Order Elliptic Problems by Mixed Finite Element Methods in Three Dimensions

where ∇ and ∇· are the gradient and divergence operators, Ω ⊂ R is an open bounded cubic domain with boundary Γ, n indicates the outward unit normal vector along Γ, A−1 = (αij)3×3 is a full positive definite matrix uniformly in Ω. Mixed finite element methods [1] should be employed to discretize the system (1.1). The main content of this paper is to present an analysis for the extrapolation of the mixed finite elements in three dimensions. The application of this approach in finite element methods was first established by Q. Lin [12]. The extrapolation method relies heavily on the existence of an asymptotic expansion for the error. The extrapolation of mixed finite element approximation in two dimensions was studied in [5]. In this paper, we study the three dimensional case. This paper is organized in the following way. In Section 2, we establish the approximation subspace and the variational formulation for the problem (1.1) and the Raviart-Thomas interpolation. The asymptotic expansion for the Raviart-Thomas interpolation is derived in Section 3. Section 4 is devoted to investigating the asymptotic expansion of the error between the mixed finite element solution and the Raviart-Thomas interpolation of the exact solution to (1.1). Based on the expansion, the asymptotic expansion of the mixed finite element approximation is demonstrated by an interpolation postprocessing method in Section 5. Hence, The extrapolation can be used to improve the accuracy of the mixed finite element solution. Some concluding remarks are given in the finial section.