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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
    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
    A particle model of compressible fluids (1997)
    Springer
    Numerische Mathematik 76 (1997), S. 111-142 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991): 76N99, 76M25, 65M12, 65M99
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary. Fluid mechanics describes the motion of mass in space under the influence of internal and external forces. The particle model presented in this article is based on this fact. The fluid is subdivided into a finite number of small mass packets, the particles. These mass packets have a finite extension and share all properties with the fluid, except for the restriction that they cannot get deformed and can perform only rigid body motions. The forces acting upon the particles are identical to those acting on a part of a fluid. The exact conservation of mass and, for the case of adiabatic flows, also of entropy is automatically guaranteed by the approach. When the particle size tends to zero, the mean local displacement of the particles converges in the weak sense. In the inviscid case, the resulting flows can be regarded as solutions of the Euler equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050256
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  • 4
    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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  • 5
    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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  • 6
    Digitale Medien
    Digitale Medien
    Die gleichmäßige Stabilität singulär gestörter diskreter und kontinuierlicher Randwertprobleme (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 223-253 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS: 65L05 ; 65L10 ; 34D15 CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: 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.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01390214
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  • 7
    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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  • 8
    Digitale Medien
    Digitale Medien
    On the multi-level splitting of finite element spaces (1986)
    Springer
    Numerische Mathematik 50 (1986), S. 123-123 
    Details
    ISSN: 0945-3245
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389672
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  • 9
    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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  • 10
    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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