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  • 1
    Electronic Resource
    Electronic Resource
    Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form (1977)
    Springer
    Numerische Mathematik 28 (1977), S. 1-14 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Conditions are given for the validity of long range quasi optimal error estimates of the Galerkin method for quasilinear parabolic equations. If the corresponding nonlinear elliptic operator satisfies a coercivity condition, the stationary problem may be approximated by the numerical solution of the parabolic problem when the time variablet is large enough, starting with an arbitrary initial function. Some practical applications are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01403853
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  • 2
    Electronic Resource
    Electronic Resource
    On the rate of convergence of the preconditioned conjugate gradient method (1986)
    Springer
    Numerische Mathematik 48 (1986), S. 499-523 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum. We present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389448
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  • 3
    Electronic Resource
    Electronic Resource
    Conditioning analysis of block incomplete factorizations and its application to elliptic equations (1997)
    Springer
    Numerische Mathematik 78 (1997), S. 189-209 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65F10, 65F15, 65F50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The paper deals with eigenvalue estimates for block incomplete factorization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial improvement of earlier results. Finally, the results are used to estimate bounds for every eigenvalue of the preconditioned matrices, in particular, for the maximum eigenvalue, when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions. The analysis yields a new upper bound of type $\gamma h^{-1}$ for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influence the positive constant $\gamma$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050310
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  • 4
    Electronic Resource
    Electronic Resource
    On the eigenvalue distribution of a class of preconditioning methods (1986)
    Springer
    Numerische Mathematik 48 (1986), S. 479-498 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F10 ; CR: G1. 3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equationAx=b, whereA is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrixA and include both pointwise and blockwise factorization. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues ofC −1 A, whereC is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389447
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  • 5
    Electronic Resource
    Electronic Resource
    A note on a class of stronglyA-stable methods (1972)
    Springer
    BIT 12 (1972), S. 1-4 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Formulae for a class ofA-stable quadrature methods, or equivalently a certain implicit Runge-Kutta scheme, are given. A short proof of the strongA-stability is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01932668
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  • 6
    Electronic Resource
    Electronic Resource
    A class ofA-stable methods (1969)
    Springer
    BIT 9 (1969), S. 185-199 
    Details
    ISSN: 1572-9125
    Keywords: Differential equations ; Quadrature method ; A-stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of methods for the numerical solution of systems of ordinary differential equations is given which—for linear systems—gives solutions which conserve the stability property of the differential equation. The methods are of a quadrature type $$y_{i,r} = y_{n,r - 1} + h\sum\limits_{k = 1}^n {a_{ik} f(y_{k,r} ), n = 1,2, \ldots ,n, r = 1,2, \ldots ,} y_{n,0} given$$ wherea ik are quadrature coefficients over the zeros ofP n −P n−1 (v=1) orP n −P n−2 (v=2), whereP n is the Legendre polynomial orthogonal on [0,1] and normalized such thatP m (1)=1. It is shown that $$\left| {y_{n,r} - y(rh) = 0(h^{2n - _v } )} \right|$$ wherey is the solution of $$\frac{{dy}}{{dt}} = f(y), t \mathbin{\lower.3ex\hbox{$\buildrel〉\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 0, y(0) given.$$
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01946812
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  • 7
    Electronic Resource
    Electronic Resource
    Global integration of differential equations through Lobatto quadrature (1964)
    Springer
    BIT 4 (1964), S. 69-86 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For the numerical solution of the initial value problemy′=f(x,y), −1≦x≦1;y(−1)=y 0 a global integration method is derived and studied. The method goes as follows. At first the system of nonlinear equations is solved. The matrix (A i,k (n) ) of quadrature coefficients is “nearly” lower left triangular and the pointsx k,n ,k=1,2,...,n are the zeros ofP n −P n−2, whereP n is the Legendre polynomial of degreen. It is showed that the errors From the valuesf(x i,n ,y i,n ),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01939850
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  • 8
    Electronic Resource
    Electronic Resource
    Stabilization of algebraic multilevel iteration methods; additive methods (1999)
    Springer
    Numerical algorithms 21 (1999), S. 23-47 
    Details
    ISSN: 1572-9265
    Keywords: multilevel method ; stabilization ; finite element method ; additive method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract There exist two main versions of preconditioners of algebraic multilevel type, the additive and the multiplicative methods. They correspond to preconditioners in block diagonal and block matrix factorized form, respectively. Both can be defined and analysed as recursive two-by-two block methods. Although the analytical framework for such methods is simple, for many finite element approximations it still permits the derivation of the strongest results, such as optimal, or nearly optimal, rate of convergence and optimal, or nearly optimal order of computational complexity, when proper recursive global orderings of node points have been used or when they are applied for hierarchical basis function finite element methods for elliptic self-adjoint equations and stabilized in a certain way. This holds for general elliptic problems of second order, independent of the regularity of the problem, including independence of discontinuities of coefficients between elements and of anisotropy. Important ingredients in the methods are a proper balance of the size of the coarse mesh to the finest mesh and a proper solver on the coarse mesh. This paper presents in a survey form the basic results of such methods and considers in particular additive methods. This method has excellent parallelization properties.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019136808500
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  • 9
    Electronic Resource
    Electronic Resource
    On the sublinear and superlinear rate of convergence of conjugate gradient methods (2000)
    Springer
    Numerical algorithms 25 (2000), S. 1-22 
    Details
    ISSN: 1572-9265
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The convergence of conjugate gradient methods for solving systems of linear equations with a symmetric positive definite matrix takes typically place in three phases, with a sublinear, linear and superlinear convergence rate, respectively. This behaviour can be explained using various generalized condition numbers which depend on all eigenvalues and on the initial error vector and using annihilating polynomials for the extreme eigenvalues. The analysis also indicates that it can be most efficient to use different preconditioners in different phases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1016694031362
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  • 10
    Electronic Resource
    Electronic Resource
    A class of preconditioned conjugate gradient methods for the solution of a mixed finite element discretization of the biharmonic operator (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 1001-1019 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The homogeneous Dirichlet problem for the biharmonic operator is solved as the variational formulation of two coupled second-order equations. The discretization by a mixed finite element model results in a set of linear equations whose coefficient matrix is sparse, symmetric but indefinite. We describe a class of preconditioned conjugate gradient methods for the numerical solution of this linear system. The precondition matrices correspond to incomplete factorizations of the coefficient matrix. The numerical results show a low computational complexity in both number of computer operations and demand of storage.
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620140705
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