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  • 1
    Digitale Medien
    Digitale Medien
    The hierarchical basis multigrid method (1988)
    Springer
    Numerische Mathematik 52 (1988), S. 427-458 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR:G1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01462238
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  • 2
    Digitale Medien
    Digitale Medien
    A class of iterative methods for solving saddle point problems (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 645-666 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10 ; 65N20 ; 65N30 ; CR: G 1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01405194
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  • 3
    Digitale Medien
    Digitale Medien
    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
    Schlagwort(e): AMS(MOS) ; 65F10 ; 65F35 ; 65N20 ; 65N30 ; CR: G 1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01385617
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  • 4
    Digitale Medien
    Digitale Medien
    A basic norm equivalence for the theory of multilevel methods (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 455-476 
    Details
    ISSN: 0945-3245
    Schlagwort(e): 65N55 ; 65N50
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01388699
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  • 5
    Digitale Medien
    Digitale Medien
    Coarse grid spaces for domains with a complicated boundary (1999)
    Springer
    Numerical algorithms 21 (1999), S. 387-392 
    Details
    ISSN: 1572-9265
    Schlagwort(e): multigrid methods ; domains of complicated shape ; 65N55 ; 65N30
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik , Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1019157313043
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