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Hierarchical Basis for the ConvectionDiffusion Equation on Unstructured Meshes
 in Ninth International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1997
"... Introduction The Hierarchical Basis Multigrid Method was originally developed for sequences of refined meshes. Hierarchical basis functions can be constructed in a straightforward fashion on such sequences of nested meshes. The HBMG iteration itself is just a block symmetric GauSeidel iteration ap ..."
Abstract

Cited by 6 (2 self)
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Introduction The Hierarchical Basis Multigrid Method was originally developed for sequences of refined meshes. Hierarchical basis functions can be constructed in a straightforward fashion on such sequences of nested meshes. The HBMG iteration itself is just a block symmetric GauSeidel iteration applied to the stiffness matrix represented in the hierarchical basis. Because the stiffness matrix is less sparse than when the standard nodal basis functions are used, the iteration is carried out by forming the hierarchical basis stiffness matrix only implicitly. The resulting algorithm is strongly connected to the classical multigrid Vcycle, except that only a subset of the unknowns on each level is smoothed during the relaxation steps [BDY88]. In recent years, we have generalized such bases to completely unstructured meshes, not just those arising from some refinement process. This is done by recognizing the strong connection between the Hierarchical Basis Multigrid Method and an
The Generalized Hierarchical Basis TwoLevel Method for the ConvectionDiffusion Equation on a Regular Grid
, 1996
"... . We make a theoretical analysis of the application of the generalized hierarchical basis multigrid method to the convectiondiffusion equation, discretized using the ScharfetterGummel discretization. Our analysis is performed for two levels of grid refinement in which we compare the effects of dif ..."
Abstract

Cited by 2 (0 self)
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. We make a theoretical analysis of the application of the generalized hierarchical basis multigrid method to the convectiondiffusion equation, discretized using the ScharfetterGummel discretization. Our analysis is performed for two levels of grid refinement in which we compare the effects of different interpolation factors for the coarse grid basis functions on the method. In particular, we find the asymptotic convergence rates for the ScharfetterGummel and the ILUfactors. The ILUfactors produce convergence rates independent of the convection directions but dependent on the size of the convection vector. Numerical results illustrating these rates are given. 1 Introduction Hierarchical basis methods define a robust class of algorithms for solving elliptic partial differential equations, especially for large systems arising in conjunction with adaptive local mesh refinement techniques. They are strongly related to classical multigrid methods, except that only a subset of the unk...