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  • CR: 5.17  (2)
  • CR: G1.8  (2)
  • Mathematics Subject Classification (1991):76N99, 76M25, 65M99  (2)
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  • 1
    Digitale Medien
    Digitale Medien
    Die maximale Konsistenzordnung von Differenzenapproximationen nichtnegativer Art (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 119-123 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N10 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Many difference methods for the numerical solution of elliptic boundary value problems lead to systems of linear equations whose matrices areM-matrices and which therefore have nonnegative inverses. In this paper it is shown, that these difference methods are at most consistent of second order.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400922
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  • 2
    Digitale Medien
    Digitale Medien
    Preconditioning indefinite discretization matrices (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 719-734 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65F10, 65N20, 65N30 ; CR: G1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396490
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  • 3
    Digitale Medien
    Digitale Medien
    On the multi-level splitting of finite element spaces (1986)
    Springer
    Numerische Mathematik 49 (1986), S. 379-412 
    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 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389538
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  • 4
    Digitale Medien
    Digitale Medien
    Die mehrstellenformeln für den Laplaceoperator (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 171-187 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Difference methods for the numerical solution of linear partial differential equations may often be improved by using a weighted right hand side instead of the original right hand side of the differential equation. Difference formulas, for which that is possible, are called “Mehrstellenformeln’ or Hermitian formulas. In this paper the Hermitian formulas for the approximation of Laplace's operator are characterized by a very simple condition. We prove, that in two-dimensional case for a Hermitian formula of ordern at leastn+3 discretization points are necessary. We give examples of such optimal formulas of arbitrary high-order.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396058
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  • 5
    Digitale Medien
    Digitale Medien
    Particles of variable size (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 143-159 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991):76N99, 76M25, 65M99
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050414
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  • 6
    Digitale Medien
    Digitale Medien
    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
    Schlagwort(e): Mathematics Subject Classification (1991):76N99, 76M25, 65M99
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050415
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