Adaptive Finite Element Algorithms for Numerical Multiscale Methods

Description

Most problems that we encounter in nature share the common feature of possessingmultiple scales. In order to modeling and approximate these problems with limitedcomputational resources, effective multiscale modeling and numerical methods areessential. Multiscale finite element method (MsFEM) and finite element heterogeneousmultiscale method (FE-HMM) are two examples of successful numerical methods basedon the homogenization theory for multiscale problems. They have been applied to manymultiscale problems successfully. For many practical problems, it is still a verychallenging task to modify these algorithms or design new algorithms to solve themultiscale problems accurately and effectively. For example, in classical MsFEM andFE-HMM, oversampling methods are often used to reduce the resonance effect. This willheavily increase the computational cost. Also, problems other than well-studied modelproblems need careful analysis and problem specific numerical multiscale methods.The long-term goal of the P.I. is to develop and study, theoretically as well ascomputationally, accurate and efficient numerical methods for solving problems withmultiscale features. In this proposal, the P.I. will concentrate on applications of ideas ofadaptive finite element methods in numerical multiscale methods including MsFEM andFE-HMM.With accurate a posteriori error estimators and carefully designed adaptive algorithms,adaptive finite element methods solve the partial differential equations with a problemadaptedmesh, thus can achieve optimal computational complexity.This feature makesadaptive algorithms ideal for numerical methods for multiscale problems.In this proposed research, the P.I. will develop all-levele adaptive numerical multiscalemethods. The adaptivity will be performed on every level of the computation, includinglocal computations of oscillating functions, deciding the size of oversampling domain,and global contributions due to geometric singularities. The P.I. will also extend themethods to other interesting problems other than standard model elliptic equations. Theproposed research will also develop new methods to combine best features of adaptivemethods and numerical multiscale methods by applying the former on non-scaleseparatedregions and the latter on scale-separated regions.