Mathematical Sciences: Theory and Applications of Multigrid Methods

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In this project, the investigator will study nonconforming multigrid methods. Robust multigrid algorithms for various finite element methods in elasticity will be developed, the performance of which will not deteriorate as the material becomes nearly incompressible. A nonconforming multigrid method for a fluid flow free-boundary problem will also be studied. Numerical experiments will be performed for these problems. In addition, some theoretical topics (conforming and nonconforming) such as pointwise convergence of multigrid methods, and multigrid methods that can overcome the pollution effects of singular solutions, will also be investigated. The results of this research will lead to efficient algorithms for the solution of large-scale computational problems in fluid mechanics and elasticity. They will find many applications in various fields of engineering and physics.