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  • Artikel  (31)
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  • 1
    Digitale Medien
    Digitale Medien
    Stability and accuracy of time discretizations for initial value problems (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 245-296 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400542
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  • 2
    Digitale Medien
    Digitale Medien
    Adams methods for neutral functional differential equations (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 221-230 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper Adams type methods for the special case of neutral functional differential equations are examined. It is shown thatk-step methods maintain orderk+1 for sufficiently small step size in a sufficiently smooth situation. However, when these methods are applied to an equation with a “non-smooth” solution the order of convergence is only one. Some computational considerations are given and numerical experiments are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01408695
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  • 3
    Digitale Medien
    Digitale Medien
    The ɛ-algorithm and multivariate Padé-approximants (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 39-46 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In the univariate case the ɛ-algorithm of Wynn is closely related to the Padé-table in the following sense: if we apply the ɛ-algorithm to the partial sums of the power series $$f(x) = \sum\limits_{i = 0}^\infty {c_i x^i } $$ then ε 2m l−m is the (l, m) Padé-approximant tof(x) wherel is the degree of the numerator andm is the degree of the denominator [1 pp. 66–68]. Several generalizations of the ɛ-algorithm exist but without any connection with a theory of Padé-approximants. Also several definitions of the Padé-approximant to a multivariate function exist, but up till now without any connection with the ɛ-algorithm. In this paper, we see that the multivariate Padé-approximants introduced in [3], satisfy the same property as the univariate Padé-approximants: if we apply the ɛ-algorithm to the partial sums of the power series $$f\left( {x_1 ,...,x_n } \right) = \sum\limits_{i_1 + ... + i_n = 0}^\infty {c_{i_1 ...i_n } x_1^{i_1 } ...x_n^{i_n } } $$ then ε 2m (l−m) is the (l, m) multivariate Padé-approximant tof(x 1, ...,x n ).
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459073
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  • 4
    Digitale Medien
    Digitale Medien
    Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 169-177 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400537
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  • 5
    Digitale Medien
    Digitale Medien
    A stable method for the evaluation of a polynomial and of a rational function of one variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 309-319 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; 65G05 ; CR: 5.11
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary It is shown that if one uses a carefully defined concept of stability (Stewart, G.W., Introduction to matrix computations, Academic Press, New York and London, 1973) then Horner's rule for the evaluation of a polynomial and some other evaluation methods are not always stable. A method is presented which is always stable. The operations count for this method is the same as that for Horner's rule. The method is generalized to apply to all rational functions of one variable.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01418328
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  • 6
    Digitale Medien
    Digitale Medien
    A semi-implicit mid-point rule for stiff systems of ordinary differential equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 373-398 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The paper introduces a new semi-implicit extrapolation method especially designed for the numerical solution of stiff systems of ordinary differential equations. The existence of a quadratic asymptotic expansion in terms of the stepsize is shown. Moreover, the new discretization is analyzed in the light of well-known stability models. The efficiency of the new integrator is clearly demonstrated by solving a series of challenging test problems including real life examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01418331
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  • 7
    Digitale Medien
    Digitale Medien
    Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 345-371 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01418330
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  • 8
    Digitale Medien
    Digitale Medien
    On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 1-13 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere $$\mathbb{C}$$ ). It is shown that the solutions of these difference equations are bounded forn≧0, uniformly for λh∈D.S is itself closed in $$\mathbb{C}$$ iff ∂S is free of cusps. The question is studed by means of contractivity analysis and a matrix theorem, derived from the matrix theorem of Kreiss.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400914
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  • 9
    Digitale Medien
    Digitale Medien
    Order and stepsize control in extrapolation methods (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 399-422 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The paper presents a new theory for joint order and stepsize control in extrapolation methods. This theory defines a locally optimal order that can be determined along any trajectory to be computed. In addition, Shannon's information theory is applied to derive some ideal convergence model that is expected to describe the behavior of an extrapolation method over a large set of test problems. Extensive numerical comparisons document a drastic acceleration in stiff integration and a mild acceleration in non-stiff integration by the new device. Moreover, a significant increase in reliability, robustness, and portability of the extrapolation codes is achieved.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01418332
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  • 10
    Digitale Medien
    Digitale Medien
    Split linear multistep methods for the numerical integration of stiff differential systems (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 299-310 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A new approach to the problem of numerically integrating stiff differential systems is described. In this approach a linear multistep method (the basic method) is split into a kind of predictor-corrector scheme, where the predictor is also implicit. If this splitting is done in an appropriate manner, the modified method has considerably better stability properties than the basic method. As a result, splitting methods are particularly useful for problems where conventional integration methods experience stability difficulties. In particular some highly stable split linear multistep methods based on backward differentiation formulae are derived and a highly stable variable step implementation is proposed.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389575
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  • 11
    Digitale Medien
    Digitale Medien
    On the integration of stiff systems of O.D.E.s using extended backward differentiation formulae (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 235-246 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A class of extended backward differentiation formulae suitable for the approximate numerical integration of stiff systems of first order ordinary differential equations is derived. An algorithm is described whereby the required solution is predicted using a conventional backward differentiation scheme and then corrected using an extended backward differentiation scheme of higher order. This approach allows us to developL-stable schemes of order up to 4 andL(α)-stable schemes of order up to 9. An algorithm based on the integration formulae derived in this paper is illustrated by some numerical examples and it is shown that it is often superior to certain existing algorithms.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396701
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  • 12
    Digitale Medien
    Digitale Medien
    A mixed finite element approximation of the Navier-Stokes equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 381-404 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We study in this paper the convergence of a new mixed finite element approximation of the Navier-Stokes equations. This approximation uses low order Lagrange elements, leads to optimal order of convergence for the velocity and the pressure, and induces an efficient numerical algorithm for the solution of this problem.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399007
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  • 13
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    Digitale Medien
    Solving complex approximation problems by semiinfinite - finite optimization techniques: A study on convergence (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 411-420 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We transform a complex approximation problem into an equivalent semiinfinite optimization problem whose constraints are described in terms of a quantity ϕ∈[0,2π[=I. We study the effect of disturbing the problem by replacingI by a compact subsetM⊂I which includes as special case the discrete case whereM consists only of finitely many points. We introduce a measure ɛ for the deviation ofM fromI and show that in any complex approximation problem the minimal distance of the disturbed problem converges quadratically with ɛ→0 to the minimal distance of the undisturbed problem which is a generalization of a result by Streit and Nuttall. We also show that in a linear finite dimensional approximation problem the convergence of the coefficients of the disturbed problem is in general at most linear. There are some graphical representations of best complex approximations computed with the described method.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01407872
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  • 14
    Digitale Medien
    Digitale Medien
    Quotienten-Differenzen-Algorithmus: Beweis der Regeln von Rutishauser (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 93-98 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.15
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Beschreibung / Inhaltsverzeichnis: Zusammenfassung Der Quotienten-Differenzen-Algorithmus nach Rutishauser ist geeignet zur Bestimmung von Polen meromorpher Funktionen, gegeben durch eine Taylorreihe. Sind mehrere Pole betragsgleich, so kann eine Polynomfolge bestimmt werden, deren Grenzpolynom diese Pole als Nullstellen hat. Die Konvergenz wurde von Rutishauser jedoch nicht bewiesen. Ein Beweis wird in der vorliegenden Arbeit präsentiert.
    Notizen: Summary The Quotient-Difference Algorithm of Rutishauser can be used for the determination of poles of a meromorphic function given by its power series. If some of the poles have same modulus, a sequence of polynomials can be determined such that the limiting polynomial has exactly these poles as zeros. The convergence has not been proved by Rutishauser, however. A proof is presented in this paper.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01459078
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  • 15
    Digitale Medien
    Digitale Medien
    Unicity of best one-sidedL 1-approximations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 229-243 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper deals with the problem of uniqueness in one-sidedL 1-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL 1-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01400541
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  • 16
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    Digitale Medien
    The finite element method for parabolic equations (1982)
    Springer
    Numerische Mathematik 40 (1982), S. 339-371 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; 65M20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this first of two papers, computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time-independent space meshes. The effectiveness of the error estimator is related to conditions on the solution regularity, mesh family type, and asymptotic range for the mesh size. For clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used. The results extend straight-forwardly to systems of equations and higher order elements in one space dimension, while the higher dimensional case requires additional considerations. The theory presented here provides the basis for the analysis and adaptive construction of time-dependent space meshes, which is the subject of the second paper. Computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396451
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  • 17
    Digitale Medien
    Digitale Medien
    Error analysis for Laplace transform —Finite element solution of hyperbolic equations (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 55-62 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The finite element method with Laplace transform of time variable is proposed for the solution of hyperbolic equations. Error estimates in Hardy spaces of functions with values in Sobolev spaces are derived. Due to the isometric isomorphism of Hardy spaces with weighted Hilbert spaces these estimates are valid also for original formulations of hyperbolic equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396305
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  • 18
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    Digitale Medien
    An analysis of thep-version of the finite element method for nearly incompressible materials (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 39-53 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; 73K25 CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper we analyze the behavior of the so-calledp-version of the finite element method when applied to the equations of plane strain linear elasticity. We establish optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[. This shows that thep-versiondoes not exhibit the degeneracy phenomenon which has led to the use of various, only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396304
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  • 19
    Digitale Medien
    Digitale Medien
    A Remez type algorithm for spline functions (1983)
    Springer
    Numerische Mathematik 41 (1983), S. 117-146 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A Remez type algorithm for computing best spline approximations of degreen withk fixed knots is developed. It is shown that the sequence constructed in the algorithm converges to a best approximation, ifk≦n+1, and at least to a nearly best approximation, ifk〉n+1. In contrast to the standard interpolation methods no restriction to the position of the knots is required. This fact may be important to treat approximation problems for splines with free knots.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396309
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  • 20
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    Digitale Medien
    On conforming finite element methods for the inhomogeneous stationary Navier-Stokes equations (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 173-194 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We consider the stationary Navier-Stokes equations, written in terms of the primitive variables, in the case where both the partial differential equations and boundary conditions are inhomogeneous. Under certain conditions on the data, the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed. A conforming finite element method is presented and optimal estimates for the error of the approximate solution are proved. In addition, the convergence properties of iterative methods for the solution of the discrete nonlinear algebraic systems resulting from the finite element algorithm are given. Numerical examples, using an efficient choice of finite element spaces, are also provided.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395310
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  • 21
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    Digitale Medien
    The QD-algorithm and multivariate Padé-approximants (1983)
    Springer
    Numerische Mathematik 42 (1983), S. 259-269 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The quotient-difference (QD) algorithm can be used to construct univariate Padé-approximants [1]. In this paper we see that it can also be used to construct the multivariate Padé-approximants introduced in [3], just by reformulating the quotient-difference algorithm as in Sect. 1. The multivariate Padé-approximants and the multivariate QD-scheme are treated in the Sects. 2 and 3 respectively. Thus for this type of multivariate Padé-approximants a link with the theory of multivariate continued fractions is established.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389572
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  • 22
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    Digitale Medien
    Strong uniqueness and second order convergence in nonlinear discrete approximation (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 439-455 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Strong uniqueness has proved to be an important condition in demonstrating the second order convergence of the generalised Gauss-Newton method for discrete nonlinear approximation problems [4]. Here we compare strong uniqueness with the multiplier condition which has also been used for this purpose. We describe strong uniqueness in terms of the local geometry of the unit ball and properties of the problem functions at the minimum point. When the norm is polyhedral we are able to give necessary and sufficient conditions for the second order convergence of the generalised Gauss-Newton algorithm.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01403680
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  • 23
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    Digitale Medien
    Unconditionally stable explicit methods for parabolic equations (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 57-68 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; 65M20 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396370
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  • 24
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    Digitale Medien
    Existence and approximation results for nonlinear mixed problems: Application to incompressible finite elasticity (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 365-382 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary This paper considers the problems of minimizing Gateaux-differentiable functionals over subsets of real Banach spaces defined by a non-linear equality constraint. The existence of a Lagrange multiplier is proved, together with approximation results on the constrained subset, provided a nonlinear compatibility condition, generalizing the classical inf-sup condition, is satisfied. These ideas are applied to equilibrium problems in incompressible finite elasticity and lead to convergence results for these problems.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396438
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  • 25
    Digitale Medien
    Digitale Medien
    The numerical solutions of interface problems by infinite element method (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 39-50 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper we derive error estimates for infinite element method used in the approximation of solutions of interface problems. Furthermore, approximations of stress intensity factors are given. The infinite element method may be considered as a certain scheme of mesh refinement, but it has the advantages that the refinement is easy to be constructed that the stiffness matrix can be calculated efficiently, and that an approximate solution which has a singularity at the singular point can be also obtained.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399310
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  • 26
    Digitale Medien
    Digitale Medien
    Optimal cycles in doubly weighted graphs and approximation of bivariate functions by univariate ones (1982)
    Springer
    Numerische Mathematik 39 (1982), S. 65-84 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65D15 ; CR: 5.13
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The problem of finding optimal cycles in a doubly weighted directed graph (Problem A) is closely related to the problem of approximating bivariate functions by the sum of two univariate functions with respect to the supremum norm (Problem B). The close relationship between Problem A and Problem B is detected by the characterization (7.4) of the distance dist (f, t) of Problem B. In Part 1 we construct an algorithm for Problem A where the essential role is played by the minimal lengthsy j(k) defined by (2.3). If weight functiont≡1 then the minimum of Problem A is computed by equality (2.4). Ift≡1 then the minimum is obtained by a binary search procedure, Algorithm 3. In Part 2 we construct our algorithms for solving Problem B by following exactly the ideas of Part 1. By Algorithm 4 we compute the minimal pseudolengthsh k(y, M) defined by (7.5). If weight functiont≡1 then the infimum dist(f,t) of Problem B is obtained by equality (7.12) which is closely related to (2.4). Ift≢1 we compute the infimum dist(f,t) by the binary search procedure Algorithm 5. Additionally, Algorithm 4 leads to a constructive proof of the existence of continuous optimal solutions of Problem B (see Theorem 7.1e) which is already known in caset≡1 but unknown in caset≢1. Interesting applications to the steady-state behaviour of industrial processes with interference (Sect. 3) and the solution of integral equations (Problem C) are included.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399312
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  • 27
    Digitale Medien
    Digitale Medien
    Analysis of finite element methods for second order boundary value problems using mesh dependent norms (1980)
    Springer
    Numerische Mathematik 34 (1980), S. 41-62 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract This paper presents a new approach to the analysis of finite element methods based onC 0-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL 2 norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL 2 norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01463997
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  • 28
    Digitale Medien
    Digitale Medien
    Convergence of the nonconforming Wilson element for arbitrary quadrilateral meshes (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 33-52 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A modified variational formulation, recently introduced by Taylor, Beresford and Wilson for solving second order problems, using the nonconforming Wilson element is here analysed. It is shown that the Patch Test is satisfied and that stresses and displacements are respectively first and second order accurate for arbitrary quadrilateral meshes.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395987
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  • 29
    Digitale Medien
    Digitale Medien
    Die Determinantenmethode zur Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 405-420 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: 5.16
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The characteristic exponent ν of the finite Hill differential equation $$y''(x) + \left( {\lambda + \sum\limits_{\kappa = 1}^k {(2t_\kappa ) \cos (2\kappa x)} } \right) y(x) = 0$$ can be evaluated from the relations $$\sin ^2 \left( {\frac{\pi }{2}v} \right) = \frac{{\pi ^2 }}{4} \det C^{(0)} \det S^{(0)}$$ or $$\cos ^2 \left( {\frac{\pi }{2}v} \right) = \det C^{(1)} \det S^{(1)} ,$$ whereS (μ) andC (μ) are certain infinite band matrices. According to Mennicken [3] the convergence of the infinite determinants can be accelerated by splitting up suitable infinite products. In the present paper this method is discussed under numerical aspects, moreover the formulas for the infinite products are simplified in such way that the complex Gamma-function is no longer needed. Finally, the presented determinental method is compared with other methods by means of some numerical examples.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01399008
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  • 30
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    Digitale Medien
    A finite element lumped mass scheme for solving eigenvalue problems of circular arches (1980)
    Springer
    Numerische Mathematik 36 (1980), S. 267-290 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper, we present a finite element lumped mass scheme for eigenvalue problems of circular arch structures, and give error estimates for the approximation. They assert that approximate eigenvalues and eigenfuctions converge to the exact ones. Some numerical examples are also given to illustrate our results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396655
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  • 31
    Digitale Medien
    Digitale Medien
    Finite dimensional approximation of nonlinear problems (1982)
    Springer
    Numerische Mathematik 38 (1982), S. 1-30 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01395805
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