The main purpose of this project is to develop, analyze, and test novel, accurate a posteriori error estimators of the recovery type for various finite element discretizations of a variety of elliptic equations and systems arising from solid and fluid mechanics, including nonlinear problems. The investigator and his colleagues plan to study two types of recovery procedures: one is accurate only for the constitutive equation and the other is accurate for both the constitutive and equilibrium equations. Based on these recovered fluxes (or the stresses for solid and fluid mechanics), they will study three kinds of estimators. In particular, 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 ideally perfect for error control (or the so-called solution verification) on coarse (pre-asymptotic) meshes. No regularity assumptions in this project mean that the only assumptions on the existence of the underlying problem are required. This is weaker than those required for approximation theory and much weaker than those required by the current theory of the recovery-based estimators. Therefore, the estimators can be applied to problems of practical interests such as interface singularities, discontinuities in the form of shock-like fronts and of interior or boundary layers. The second part of the project is to establish convergence of adaptive finite element methods based on the recovery-based estimators and the newly developed estimators of this project.
A major problem with computer simulations of physical phenomena is that all computational results obtained involve numerical error. Discretization error can be large, pervasive, unpredictable by classical heuristic means, and can invalidate numerical predictions. A posteriori error estimation is a rigorous mathematical theory for estimating and quantifying discretization error in terms of the error's magnitude and distribution based on the current simulation and given data of the underlying problem. This information provides bases for solution verification and for adaptive control of simulation process: adaptive mesh refinement, adaptive control of mathematical models and numerical algorithms. Success in this project will provide accurate and reliable a posteriori error estimators for a large class of elliptic equations/systems arising from engineering, physics, aerodynamics, atmospheric sciences, geology, biomechanics, material sciences, nano-technology, and industrial applications. The development of the exact estimator will enable error control on pre-asymptotic meshes and predictable computation analysis. Error control on pre-asymptotic meshes is of paramount importance for simulating physical phenomena in engineering applications and scientific predictions with limited computer resources.
