Robust a Posteriori Error Estimations for Various Finite Element Methods on Interface and Diffusion Problems
A posteriori error estimation has been intensely studied in the academia and widely adapted to the industry. During the past two decades, the construction, analysis, and implementation of robust a posteriori error estimators for various finite element approximations to partial differential equations with parameters have been one of the research focuses. This work studies the robust a posteriori error estimation for various finite element approximations of interface and diffusion problems with possible high jumps across interfaces. In particular, we study the robust residual-based, recovery-based, and equilibrated error estimations for conforming, nonconforming, and discontinuous Galerkin finite element approximations of interface or diffusion problems. Here robustness means that constants involved in the error estimation are independent of the jump of the coefficient. It is crucial that an a posteriori error estimation is robust for problems with discontinuous coefficients for it guarantees the optimal decay of the global error. Existing residual-based robust error estimations may be proved to be robust under the assumption that the distribution of the coefficient is quasi-monotone [17,74]. The assumption is quite restrictive in practice while many numerical experiments including ours suggest that such assumption is not necessary. In Chapter 2, we study the residual-based a posteriori error estimation for the Crouzeix-Raviart nonconforming finite element approximation to the interface problem, aiming to prove the robustness without the QMA assumption. To do so, we introduce a new and direct approach, without using the Helmholtz decomposition, to analyze the reliability of the estimator. It is proved that a slightly modified estimator is reliable with the constant independent of the jump of the coefficient, without the assumption that the diffusion coefficient is quasi-monotone. For elliptic interface problems in two and three dimensions, Chapter 3 studies the a priori and residual-based a posteriori error estimations for the Crouzeix-Raviart nonconforming and the discontinuous Galerkin finite element approximations. It is shown that both the a priori and the a posteriori error estimates are robust with respect to the diffusion coefficient, i.e., constants in the error bounds are independent of the jump of the diffusion coefficient. The a priori estimates are also optimal with respect to local regularity of the solution. Moreover, we obtained these estimates with no assumption on the distribution of the diffusion coefficient. In , we introduced and analyzed an improved Zienkiewicz-Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise "constant'' flux recovery in the H(div;Ω) conforming finite element space. Chapter 4 extends the results of  to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear H(div) space. Chapter 5 studies an equilibrated a posteriori error estimator for nonconforming finite element approximations to diffusion problems with possibly discontinuous coefficients. The error estimator is based on a newly developed inequality (see Theorem 5.3.1 or Corollary 5.3.1) that may be regarded as an extension of the Prager-Synge identity to piecewise H1(Ω) functions including nonconforming and discontinuous elements. For nonconforming finite element approximations of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrate flux in the H(div;Ω) through a local element-wise scheme and that recovers a gradient in the H(curl;Ω) through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution. The numerical results are presented at the end of Chapters 2, 4, and 5 to illustrate the performance of the error estimation.
Cai, Purdue University.
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