Adaptive Finite Element Method for Interface Problems

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The objective of the project is to develop and to analyze a robust adaptive finite element method for the Darcy-Stokes-Brinkman model with extremely variable or discontinuous parameters. The proposed research include: developing a robust adaptive H(div) finite element method; conducting a priori error analysis; deriving efficiency and reliability bounds for a posteriori error estimators; proving convergence of adaptive mesh refinement procedures; developing a computer program for implementation of the method.The Darcy-Stokes-Brinkman model has wide range of applications such as surface and sub-surface water interaction in geoscience, blood circulation in health science and, fuel cells, filtration problems in environment science. Solving these problems has attracted a lot of attention from mathematicians and engineers, and many works have been done in developing and analyzing numerical algorithms for Darcy-Stokes-Brinkman model. Majority of these works treat viscosity and permeability as either constant or near zero jump. However, for the practical relevant problems, these physical parameters are either discontinuous or highly variable. Singularity caused by the highly varied physical parameters creates tremendous difficulty in development of numerical algorithms. There is a great interest in obtaining efficient and robust numerical methods to simulate these real word problems.