A posteriori error estimate for the mixed finite element method

A computable error bound for mixed finite element methods is established in the model case of the Poisson–problem to control the error in the H(div,Ω) ×L2(Ω)–norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart–Thomas, Brezzi-Douglas-Marini, and Brezzi-DouglasFortin-Marini elements. 1. Mixed method for the Poisson problem Mixed finite element methods are well-established in the numerical treatment of partial differential equations as regards a priori error estimates to guarantee convergence [BF]. In practical applications, a posteriori error control is at least of the same importance to guarantee a reliable approximation. Moreover, a posteriori error estimators indicate adaptive mesh-refinement criteria [EEHJ, V1] for an efficient computation. In this paper we establish an efficient and reliable error estimator for the model example in the mixed finite element methods: Given f ∈ L(Ω), the Poisson problem consists in finding a function u ∈ H 0 (Ω) that satisfies div(A∇u) + f = 0 in Ω. (1.1) Here, A ∈ L∞(Ω;R2×2) is symmetric and uniformly elliptic, Ω is a convex bounded domain in the plane with polygonal boundary Γ. The Lebesgue and Sobolev spaces L(Ω) and H 0 (Ω) are defined as usual (e.g., as in [H, LM]). We assume below that (1.1) is H–regular which, according to Ω being convex, means certain regularity on A (A the unit matrix as for the Laplace equation is clearly sufficient). The mixed formulation is given by splitting (1.1) into two equations where u ∈ H 0 (Ω) and p ∈ L(Ω) are unknown and have to satisfy div p+ f = 0 and p = A∇u in Ω. (1.2) It is well-known that (1.2) has a solution (p, u) ∈ H(div,Ω)×L(Ω), where, as usual, H(div,Ω) := {q ∈ L(Ω) : div q ∈ L(Ω)} is endowed with the norm given by ‖q‖H(div,Ω) := ∫ Ω (|q| + | div q|) dx (q ∈ H(div,Ω)). Received by the editor September 12, 1995 and, in revised form, May 1, 1996. 1991 Mathematics Subject Classification. Primary 65N30, 65R20, 73C50.