The main aim of this proposed research is to design, analyze, and test efficient, reliable, and robust a posteriori error estimators for various finite element approximations to computationally challenging problems; that is, those problems having the following phenomena: interface singularities, discontinuities (in the form of shock-like fronts, and of interior and boundary layers), and/or oscillations of various scales (multiscale phenomena). Estimators to be developed in this project do not use a priori knowledge of locations and characteristics of these phenomena; they may then be applied more readily to highly nonlinear problems and have potential to be applied to complex systems arising in applications. Estimators to be investigated in this project are of the recovery type; recovery estimators possess a number of attractive features that have led to their widespread adoption in engineering practice and to the subject of mathematical study. However, existing recovery estimators have several major drawbacks for more challenging problems. The investigator and his colleagues overcome those obstacles by introducing an innovative recovery procedure and developing a methodology on how to design efficient, reliable, and robust recovery estimators for complex systems. The methodology is applied to various problems arising from continuum mechanics to design robust estimators that will be studied theoretically and numerically.
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 necessary tool in computer simulations of complex natural and engineering problems. As identified by the US National Research Council, AMR is one of two necessary tools (AMR and Parallel Computer) for computationally grand challenging problems. The key ingredient for success of AMR algorithms are a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Success in this project will empower the ability of AMR algorithms for automatically locating physical interfaces, detecting layers and discontinuities, and resolving oscillations of various scales. The methodology developed in the project will shed light on how to design estimators for indefinite problems such as convection-dominant diffusion problems on relatively coarse meshes. Research on those indefinite problems is completely open and requires major breakthrough not only technically but also conceptually.
