A Posteriori Error Estimation through Duality and Some Other Topics

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Self-adaptive numerical methods provide a powerful and automatic approach in scientific computing. In particular, Adaptive Mesh Refinement (AMR) algorithms have been widely used in computational science and engineering and have become a common tool in computer simulations of complex natural science and engineering problems. As identified by the US National Research Council, AMR is one of two necessary tools (AMR and Parallel Computing) for computationally tackling Grand Challenge problems. The key ingredient for success of AMR algorithms is a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Another challenge in computer simulations of complex systems is the reliability of computer predictions. These considerations (efficiency in AMR algorithms and error control) demonstrate the need for an error estimator that can a posteriori be extracted from the computed numerical solution and the given data of the underlying problem. Such an a posteriori error estimate ideally should provide an underlying rigorous mathematical theory for estimating and quantifying discretization error in terms of the error's magnitude and its spatial distribution. Success in this project will allow AMR algorithms to automatically locate physical interfaces, detect layers and discontinuities, and resolve oscillations of various scales. The dual estimators to be developed in this project will resolve the most natural but extremely difficult question of discretization error control on coarse meshes for a class of problems and hence partially guarantee reliability of computer simulations.This research project focuses on the development, analysis, and test of a posteriori error estimators through the methodology of duality. The dual estimators to be developed in this project will have a guaranteed reliability bound with reliability constant being one. Hence, these estimators are perfect for discretization error control and may be used as an accurate stopping criterion for iterative solvers. The methodology of duality may be applied to a large class of problems arising from continuum mechanics including linear and nonlinear problems. Since these estimators will not use a priori knowledge on the locations and characteristics of interface singularities, discontinuities (in the form of shock-like fronts, and of interior and boundary layers), and/or oscillations of various scales (multi-scale phenomena), they may then be applied more readily to highly nonlinear problems and have the potential of being applied to complex systems arising in applications. The emphases and the difficulties of the proposed research are (1) explicit or local construction of an approximation to the dual variable such that the resulting indicator is efficient and robust, and (2) theoretical and numerical confirmation of the efficiency and robustness. Finally, a small portion of the proposed research addresses an open theoretical question on the robustness of estimators for interface problems.