RUI: Investigation of Discontinuous Galerkin Least-Squares Finite Element Methods for Singularly Perturbed Problems

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The research objective of this project is to investigate discontinuousGalerkin least-squares finite element methods (DG-LS FEMs) for reaction-diffusion problems with singular perturbations. The numerical approximation of singularly perturbed problems is a practically important but still difficult subject. The analytical solutions to such problems typically contain boundary or interior layers, which cause nonphysical excessive numerical oscillations in the vicinity of layers in the solutions by standard finite element method and finite difference method. Many stabilization techniques have been developed to improve numerical solutions. The drawback of these techniques is the presence of problem-dependent parameters that need to be properly tuned according to a priori knowledge of layers, which nonetheless is not achievable in most complex problems. The DG-LS FEM is robust and efficient, which does not involve such problem-dependent parameters. The proposed research will advance knowledge and understanding of the DG-LS FEM as well as singular perturbation problems, provide a theoretical framework for the analysis of DG-LS FEMs, and establish a working principle of efficient and high quality adaptive schemes. Singularly perturbed problems have attracted a lot of attention from engineers, mathematicians and scientists because of a wide range of important applications such as fluid dynamics, electromagnetism, semiconductor research, chemotaxis, genetics, and computational biology. Numerical approximations are usually the only way for solving such problems. There is an urgent need for a problem-dependent-parameter-free numerical method for singularly perturbed problems. This project is to develop creative solutions to fulfill this need, which will lead to reliable and efficient numerical approaches for solving complex singularly perturbed reaction-diffusion problems.