Multiscale, Nonlinear and Adaptive Approximation

pp 599-659


Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

  • Jinchao XuAffiliated withDepartment of Mathematics, Pennsylvania State UniversityLMAM, The School of Mathematical Sciences, Peking University Email author 
  • , Long ChenAffiliated withDepartment of Mathematics, University of California at Irvine
  • , Ricardo H. NochettoAffiliated withDepartment of Mathematics and Institute for Physical Science and Technology, University of Maryland

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We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids.