Robust and Explicit a posteriori Error Estimation Techniques in Adaptive Finite Element Method

Difeng Cai, Purdue University

Abstract

We present a comprehensive study of robust a posteriori error estimation for finite element approximations to elliptic partial differential equations. New results are presented for two types of estimators: the hybrid estimator and the equilibrated estimator. The framework of the hybrid estimator is introduced for two classes of problems: diffusion equations with discontinuous coefficients and stationary convectiondiffusion-reaction equations with dominated convection/reaction. The hybrid estimator inherits all advantages of the residual estimator - explicit, general, robust- and is shown to be much more accurate according to extensive numerical experiments. Another kind of estimator - the flux equilibration-based estimator (or “equilibrated estimator” for simplicity) - is a second focus of the thesis. This kind of estimator is perfect for error control as it yields a guaranteed upper bound of the true error. We investiage equilibrated estimators for conforming, nonconforming, and discontinuous Galerkin discretizations of diffusion equations with discontinuous coefficients. For conforming elements, we present first a general procedure in both two and three dimensions for computing an equilibrated estimator via local minimizations and prove its robustness. We then propose an explicit equilibrated estimator for two dimensional problems and prove the robustness. This is the first robust equilibrated estimator that is explicit for conforming elements. Numerical results show that the new explicit estimator requires significantly less computational time than the state-of-the-art local minimization-based estimator. For nonconforming elements, we introduce an explicit equilibrated estimator for arbitrary order discretizations. The new estimator is robust and is independent of the odd/even order of the nonconforming finite element. This is the first robust equilibrated estimator that enjoys the degree-independent property for nonconforming elements. Besides the design of error estimators, we discuss the quasi-monotonicity condition, a central assumption for deriving robust a posteriori error estimates for interface problems. We construct an example to show that the assumption is not only sufficient but also necessary for the robustness of equilibrated estimators. All results in the thesis hold for arbitrary order finite element approximations and for both two and three dimensional problems (unless otherwise stated).

Degree

Ph.D.

Advisors

Cai, Purdue University.

Subject Area

Mathematics

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