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  • 1
    Electronic Resource
    Electronic Resource
    The hierarchical basis multigrid method for convection-diffusion equations (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 7-37 
    Details
    ISSN: 0945-3245
    Keywords: 65N30 ; 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We make a theoretical study of the application of a standard hierarchical basis multigrid iteration to the convection diffusion equation, discretized using an upwind finite element discretizations. We show behavior that in some respects is similar to the symmetric positive definite case, but in other respects is markedly different. In particular, we find the rate of convergence depends significantly on parameters which measure the strength of the upwinding, and the size of the convection term. Numerical calculations illustrating some of these effects are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385495
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  • 2
    Electronic Resource
    Electronic Resource
    An algorithm for coarsening unstructured meshes (1996)
    Springer
    Numerische Mathematik 73 (1996), S. 1-36 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F10, 65N20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order $J^2$ , where $J$ is the number of hierarchical basis levels.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050181
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  • 3
    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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  • 4
    Electronic Resource
    Electronic Resource
    Some upwinding techniques for finite element approximations of convection-diffusion equations (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 185-202 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65N05 ; 65N10 ; 65N20 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A uniform framework for the study of upwinding schemes is developed. The standard finite element Galerkin discretization is chosen as the reference discretization, and differences between other discretization schemes and the reference are written as artificial diffusion terms. These artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices. Three basis matrices are symmetric, rank one diffusion operators associated with the edges of the triangle; the fourth basis matrix is skew symmetric and is associated with a rotation by ϕ/2. While finite volume discretizations may be written as upwinded Galerkin methods, the converse does not appear to be true. Our approach is used to examine several upwinding schemes, including the streamline diffusion method, the box method, the Scharfetter-Gummel discretization, and a divergence-free scheme.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385618
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  • 5
    Electronic Resource
    Electronic Resource
    Multilevel ILU decomposition (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 543-576 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F10, 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050430
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  • 6
    Electronic Resource
    Electronic Resource
    Efficient algorithms for solving tensor product finite element equations (1978)
    Springer
    Numerische Mathematik 31 (1978), S. 49-61 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65N20 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary There are currently several highly efficient methods for solving linear systems associated with finite difference approximations of Poisson's equation in rectangular regions. These techniques are employed to develop both direct and iterative methods for solving the linear systems arising from the use ofC 0 quadratic orC 1 cubic tensor product finite elements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396013
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    An analysis of the composite step biconjugate gradient method (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 295-319 
    Details
    ISSN: 0945-3245
    Keywords: 65N20 ; 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The composite step biconjugate gradient method (CSBCG) is a simple modification of the standard biconjugate gradient algorithm (BCG) which smooths the sometimes erratic convergence of BCG by computing only a subset of the iterates. We show that 2×2 composite steps can cure breakdowns in the biconjugate gradient method caused by (near) singularity of principal submatrices of the tridiagonal matrix generated by the underlying Lanczos process. We also prove a “best approximation” result for the method. Some numerical illustrations showing the effect of roundoff error are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385699
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  • 8
    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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  • 9
    Electronic Resource
    Electronic Resource
    The Finite Volume Scharfetter-Gummel method for steady convection diffusion equations (1998)
    Springer
    Computing and visualization in science 1 (1998), S. 123-136 
    Details
    ISSN: 1433-0369
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. A priori error estimates are given for the Finite Volume Scharfetter-Gummel (FVSG) discretization of the steady convection diffusion equation by showing that the FVSG method gives the same discretization as the Edge Averaged Finite Element method of Markowich and Zlámal [9] and Xu and Zikatanov [14]. The analysis also suggests a class of modifications for triangulations containing obtuse angles. Numerical results comparing the FVSG method and a modified FVSG method to other discretizations are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s007910050012
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  • 10
    Electronic Resource
    Electronic Resource
    Sparse matrix multiplication package (SMMP) (1993)
    Springer
    Advances in computational mathematics 1 (1993), S. 127-137 
    Details
    ISSN: 1572-9044
    Keywords: Sparse matrices ; matrix multiplication ; transposition ; numerical linear algebra ; Numerical analysis ; numerical linear algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Routines callable fromFortran and C are described which implement matrix-matrix multiplication and transposition for a variety of sparse matrix formats. Conversion routines between various formats are provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02070824
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    Paper (German National Licenses)
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