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  • 1
    Electronic Resource
    Electronic Resource
    A basic norm equivalence for the theory of multilevel methods (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 455-476 
    Details
    ISSN: 0945-3245
    Keywords: 65N55 ; 65N50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Subspace decompositions of finite element spaces based onL 2-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors have proved the equivalence of the associated discrete norms with theH 1-norm. The present paper gives an elementary, self-contained derivation of this result which is based on the use ofK-functionals known from the theory of interpolation spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388699
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    The hierarchical basis multigrid method (1988)
    Springer
    Numerische Mathematik 52 (1988), S. 427-458 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR:G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß-Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigridV-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01462238
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Two preconditioners based on the multi-level splitting of finite element spaces (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 163-184 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; exclusively local properties like the shape regularity of the finite elements are utilized.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385617
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Coarse grid spaces for domains with a complicated boundary (1999)
    Springer
    Numerical algorithms 21 (1999), S. 387-392 
    Details
    ISSN: 1572-9265
    Keywords: multigrid methods ; domains of complicated shape ; 65N55 ; 65N30
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is shown that, with homogeneous Dirichlet boundary conditions, the condition number of finite element discretization matrices remains uniformly bounded independent of the size of the boundary elements provided that the size of the elements increases with their distance to the boundary. This fact allows the construction of simple multigrid methods of optimal complexity for domains of nearly arbitrary shape.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019157313043
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    Paper (German National Licenses)
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