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  • 65N30  (4)
  • CR: G1.8  (2)
  • Mathematics Subject Classification (1991):76N99, 76M25, 65M99  (2)
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Material
Years
Keywords
  • 1
    Electronic Resource
    Electronic Resource
    Preconditioning indefinite discretization matrices (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 719-734 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10, 65N20, 65N30 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can also be used for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396490
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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
    A class of iterative methods for solving saddle point problems (1989)
    Springer
    Numerische Mathematik 56 (1989), S. 645-666 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; 65N20 ; 65N30 ; CR: G 1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01405194
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    On the multi-level splitting of finite element spaces (1986)
    Springer
    Numerische Mathematik 49 (1986), S. 379-412 
    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 In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log κ)2) where κ is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like $$O\left( {\left( {\log \frac{1}{h}} \right)^2 } \right)$$ instead of $$O\left( {\left( {\frac{1}{h}} \right)^2 } \right)$$ for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389538
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    Paper (German National Licenses)
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  • 5
    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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  • 6
    Electronic Resource
    Electronic Resource
    Particles of variable size (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 143-159 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):76N99, 76M25, 65M99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. A problem of all particle methods is that they produce large vacuum regions when they are applied to a free gas flow, for example. With the approach recently proposed by the author [Numer. Math. (1997) 76: 111–142], this difficulty can be avoided. One can let the particles adapt their size to the local state of the fluid. How, is described in the present article. The diameter as an additional degree of freedom strongly improves the performance of the numerical methods based on this particle model.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050414
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Entropy generation and shock resolution in the particle model of compressible fluids (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 161-177 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):76N99, 76M25, 65M99
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. The examination of the particle model of compressible fluids that has been developed by the author [Numer. Math. (1997) 76: 111–142] and that has recently been extended to particles of variable size [Numer. Math. (1999) 82: 143–159], is continued. It is shown that, in the limit of particle sizes tending to zero, both the mass density and the mass flux density and the entropy density and the entropy flux density converge in the weak sense and satisfy the corresponding conservation laws. To incorporate entropy generation in shocks, a new kind of viscous force is introduced.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050415
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    Paper (German National Licenses)
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  • 8
    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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