Flux Recovery, A Posteriori Error Estimation, and Adaptive Finite Element Method

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The main purpose of this project is to develop, analyze, and test novel,accurate a posteriori error estimators of the recovery type for variousfinite element discretizations of a variety of elliptic equations andsystems arising from solid and fluid mechanics, including nonlinearproblems. The investigator and his colleagues plan to study two types ofrecovery procedures: one is accurate only for the constitutive equationand the other is accurate for both the constitutive and equilibriumequations. Based on these recovered fluxes (or the stresses for solidand fluid mechanics), they will study three kinds of estimators. Inparticular, they will study an exact estimator on any given mesh,including an arbitrary initial mesh, with no regularity assumptions.Exactness on any given mesh implies that the estimator is ideallyperfect for error control (or the so-called solution verification) oncoarse (pre-asymptotic) meshes. No regularity assumptions in thisproject mean that the only assumptions on the existence of theunderlying problem are required.This is weaker than those required for approximation theory and muchweaker than those required by the current theory of the recovery-basedestimators. Therefore, the estimators can be applied to problems ofpractical interests such as interface singularities, discontinuities inthe form of shock-like fronts and of interior or boundary layers. Thesecond part of the project is to establish convergence of adaptivefinite element methods based on the recovery-based estimators and thenewly developed estimators of this project.A major problem with computer simulations of physical phenomena is thatall computational results obtained involve numerical error.Discretization error can be large, pervasive, unpredictable by classicalheuristic means, and can invalidate numerical predictions.A posteriori error estimation is a rigorous mathematical theory forestimating and quantifying discretization error in terms of the error'smagnitude and distribution based on the current simulation and givendata of the underlying problem. This information provides bases forsolution verification and for adaptive control of simulation process:adaptive mesh refinement, adaptive control of mathematical models andnumerical algorithms. Success in this project will provide accurate andreliable a posteriori error estimators for a large class of ellipticequations/systems arising from engineering, physics, aerodynamics,atmospheric sciences, geology, biomechanics, material sciences,nano-technology, and industrial applications. The development of theexact estimator will enable error control on pre-asymptotic meshes andpredictable computation analysis. Error control on pre-asymptotic meshesis of paramount importance for simulating physical phenomena inengineering applications and scientific predictions with limitedcomputer resources.