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  • ddc:000  (6)
  • Mathematics Subject Classification (1991):76N99, 76M25, 65M99  (2)
  • 65L10  (1)
  • AMS(MOS)  (1)
Source
Material
Years
Keywords
Language
  • 1
    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)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Die gleichmäßige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 223-253 
    Details
    ISSN: 0945-3245
    Keywords: AMS: 65L05 ; 65L10 ; 34D15 CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper we consider singular perturbation problems for ordinary differential operators of ordern and their discrete counterparts on arbitrary nonuniform grids. We prove that the singularly perturbed initial value problem is stable uniformly in the perturbation parameter ε in both the continuous and the discrete case. We use this result to characterize the stability of the corresponding continuous and discrete boundary value problems. If the continuous problem is stable and if the consistency error is smaller than a certain constant, the discrete problem is also stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390214
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    Paper (German National Licenses)
    Fulltext
  • 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)
    Fulltext
  • 4
    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)
    Fulltext
  • 5
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    Unknown
    The Hierarchical Basis Multigrid Method. (1987)
    Details
    Publication Date: 2014-02-26
    Description: 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 multigrid V-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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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    OPUS
  • 6
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    Preconditioning Indefinite Discretization Matrices. (1987)
    Details
    Publication Date: 2014-02-26
    Description: 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 be used also 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. {\bf Keywords: }Preconditioned conjugate gradient methods, finite elements. {\bf Subject Classification: } AMS(MOS):65F10, 65N20, 65N30.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
  • 7
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    Concepts of an Adaptive Hierarchical Finite Element Code. (1988)
    Details
    Publication Date: 2014-02-26
    Description: The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device - which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved - independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local - making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem. {\bf Keywords:} finite elements, hierarchical basis, adaptive mesh refinement, preconditioned conjugate gradient methods.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
  • 8
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    Two Preconditioners Based on the Multi-Level Splitting of Finite Element Spaces. (1989)
    Details
    Publication Date: 2014-02-26
    Description: 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; our theory is based exclusively on local properties like the shape regularity of the finite elements.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
  • 9
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    A Basic Norm Equivalence for the Theory of Multilevel Methods. (1992)
    Details
    Publication Date: 2014-02-26
    Description: Subspace decompositions of finite element spaces based on $L2$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the $H^1$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of $ K$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
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    OPUS
  • 10
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    Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid. (1993)
    Details
    Publication Date: 2014-02-26
    Description: Elliptic boundary value problems are frequently posed on complicated domains which cannot be covered by a simple coarse initial grid as it is needed for multigrid like iterative methods. In the present article, this problem is resolved for selfadjoint second order problems and Dirichlet boundary conditions. The idea is to construct appropriate subspace decompositions of the corresponding finite element spaces by way of an embedding of the domain under consideration into a simpler domain like a square or a cube. Then the general theory of subspace correction methods can be applied.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
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