Numerical Study on A Posteriori Error Estimators of Gradient/Flux Recovery Type

This paper presents a numerical study on a posteriori error estimators for secondorder scalar elliptic partial differential equations. The study involves five error estimators of the recovery type. Two of them are the existing recovery based estimators based on the discrete (Zienkiewicz-Zhu estimator) and continuous L2 projections of the directly computed gradient onto the continuous piecewise linear finite element spaces. The other three are our newly developed flux recovery based error estimators. The flux is recovered in the H(div)-conforming finite element space to accommodate possible discontinuity of the flux. For several test problems, these estimators are compared in the effective index and in the optimality of the generated meshes. For non-smooth problems, our numerical results show that newly developed flux error estimators are favorable. 1 Gradient/Flux Recovery Based Error Estimators The theory of a posteriori error estimation has become an important area of research and has found application in an increasing number of commercial software products and scientific programs. The purpose of a posteriori error estimators is twofold: (1) generating an optimal mesh in a sense that a prescribed error can be achieved by solving the discrete problem with minimal grid points and (2) providing a good stopping criterion, i.e., the ratio of the error estimator to the true error in a specific norm is close to 1. We will use these two criteria to compare the error estimators for several test problems in this paper. The problem under consideration is the 2nd order elliptic equation. Let Ω be a bounded, open, connected subset of < with a Lipschitz continuous boundary ∂Ω. Consider the following second-order elliptic boundary value problem −∇ · (A∇u) + bu = f in Ω with u = 0 on ∂Ω, (1) where A is a uniformly symmetric positive definite matrix, b is a given constant, and f is a given scalar-valued function, respectively. Preprint submitted to Elsevier 7 January 2008 Let Th be a decomposition of Ω into admissible and shape-regular triangulations. Let P1(K) be the space of linear function on a triangle K, S1 := {u ∈ H(Ω) : u|K ∈ P1(K)} be the piecewise continuous linear finite element space, and S 1 = S1 ⋂ H 0 (Ω). We denote the finite element solution of the above equation in S 1 by uh. Estimators of the recovery type possess a number of attractive features that have led to their popularity. In particular, their ease of implementation, generality, and ability to produce quite accurate estimators on fine meshes have led to their widespread adoption, especially in the engineering community. However, these estimators have some drawbacks. First, they are not reliable on coarse meshes. Second, for applications with non-smooth solution, they can possibly overrefine regions where there are no error. Third, there is a gap between theory and practice. Here are two existing estimators of recovery type. 1 ZZ. (Zienkiewicz-Zhu) error estimator [Zienkiewicz and Zhu. Internat. J. Numer. Methods Engrg., 33 (1992),1331-1364 and 1365-1382]. The first and popular recovery based estimator is the Zienkiewicz-Zhu error estimator. Let G(uh) ∈ S 1 be a post-processed approximation of ∇uh by some averaging technique (see Zienkiewicz and Zhu’s paper). Then the ZZ estimator is defined by ηZZ = ‖G(uh)−∇uh‖0,Ω. 2 L2S1. Error estimator based on the L projection onto S 1 space [Carstensen and Bartels, Math. Comp., 71(239) (2002), 945-969]. Let σh ∈ S 1 be the solution of (σh, τ ) = (∇uh, τ ), for all τ ∈ S 1 , then ηL2S1 is defined by ηL2S1 = ‖σ −∇uh‖0,Ω. Recently, we developed and analyzed three estimators based on the recovery of the flux σ = −A∇u. To eliminate possible over-refinement, we recover the flux in the H(div) conforming finite element spaces like Raviart-Thomas and Brezzi-Douglas-Marini elements, that accommodate possible discontinuities of the tangential components of the flux. More specifically, let RT0 be the lowest order Raviart-Thomas H(div) conforming space [Raviart and Thomas, Lect. Notes Math. 606, Springer-Verlag, Berlin and New York (1977), 292-315.]. Let σl2 ∈ RT0 and σhdiv ∈ RT0 be the solutions of the following problems (Aσl2, τ )=−(∇uh, τ ) ∀ τ ∈ RT0, (Aσhdiv, τ ) + (∇ · σhdiv,∇ · τ )=−(∇uh, τ ) + (f − uh,∇ · τ ) ∀ τ ∈ RT0, respectively. Based on these two recoveries of the flux, we studied two types of estimators: (i) the weighted L norm of difference between the direct and post-processed approximations of the flux; and (ii) the estimator in (i) plus the L norm of the element residual. Adding the element residual makes the recovery based estimators reliable.