Robust recovery based a posteriori error estimators for various lower-order finite element approximations to interface problems
Abstract
The a posteriori error estimators of the recovery type are extremely popular in the engineering community due to their many appealing properties: simplicity, universality, and asymptotic exactness. However, these estimators have a major drawback. For applications with non-smooth solution like elliptic interface equations, they over-refine regions where there is no error. The reason of the over-refinement of existing recovery based error estimators is simply that continuous functions (recovered gradient/flux functions) are used to approximate discontinuous functions (true gradient/flux functions) in the recovery procedure. In this work, we study robust recovery based error estimators for finite element approximations to the interface problem, which is the diffusion equation with discontinuous coefficients. By using the intrinsic continuities of the underlying problem and the properties of different finite element discretizations, we present a unifying framework for constructing robust recovery based error estimators for various finite element approximations of the lowest-order, i.e., conforming, mixed, nonconforming, and discontinuous Galerkin finite element methods. The guideline for choices of the quantities to be recovered and the proper finite element spaces to be used is based on our view that the recovery estimators measure the violation of finite element approximations on physical continuities. Therefore, the quantity to be recovered is that whose finite element approximation does not preserve the physical continuity. For diffusion problems, it is then the flux or/and the gradient depending on the discretization schemes, and the proper finite element spaces is the conforming finite elements of H(div) for the flux and of H(curl) for the gradient. By doing this, we develop robust recovery based error estimators, whose reliability and efficiency bounds are proved to be independent of the jumps of coefficients. These estimators do not over-refine regions where there is no error. Finally, numerical results are presented to support the theoretical analysis.
Degree
Ph.D.
Advisors
Cai, Purdue University.
Subject Area
Mathematics
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