Titlebook: Domain Decomposition Methods in Science and Engineering XX; Randolph Bank,Michael Holst,Jinchao Xu Conference proceedings 2013 Springer-Ve

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Mesh Regularization in Bank-Holst Parallel ,-Adaptive MeshingIn this work, we study mesh regularization in Bank-Holst parallel adaptive paradigm when adaptive enrichment in both . (geometry) and . (degree) is used. The paradigm was first introduced by Bank and Holst in [1–3] and later extended to .-adaptivity in [5]. In short, the paradigm can be summarized in the following steps.

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Multigrid Methods for the Biharmonic Problem with Cahn-Hilliard Boundary ConditionsLet . be a bounded polygonal domain, . and .

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Shifted Laplacian RAS Solvers for the Helmholtz EquationWe consider the Helmholtz equation: . where . is a bounded polygonal region in k., and the . correspond to subsets of . where the Dirichlet, Neumann and Sommerfeld boundary conditions are imposed.

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Equidistribution and Optimal Approximation Classension. In recent years, mathematicians start to prove the convergence and optimal complexity of the adaptive procedure in multi-dimensions. Dörfler [11] first proved an error reduction in the energy norm for the Poisson equation provided the initial mesh is fine enough.

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