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  • 1
    Electronic Resource
    Electronic Resource
    A posteriori error estimates for mixed FEM in elasticity (1998)
    Springer
    Numerische Mathematik 81 (1998), S. 187-209 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65N30, 65N15, 73C35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lamé constant $\lambda$ , and so locking phenomena for $\lambda\to\infty$ are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050389
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  • 2
    Electronic Resource
    Electronic Resource
    Numerical analysis of the primal problem of elastoplasticity with hardening (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 577-597 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N30, 65R20, 73C50
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The finite element method is a reasonable and frequently utilised tool for the spatial discretization within one time-step in an elastoplastic evolution problem. In this paper, we analyse the finite element discretization and prove a priori and a posteriori error estimates for variational inequalities corresponding to the primal formulation of (Hencky) plasticity. The finite element method of lowest order consists in minimising a convex function on a subspace of continuous piecewise linear resp. piecewise constant trial functions. An a priori error estimate is established for the fully-discrete method which shows linear convergence as the mesh-size tends to zero, provided the exact displacement field u is smooth. Near the boundary of the plastic domain, which is unknown a priori, it is most likely that u is non-smooth. In this situation, automatic mesh-refinement strategies are believed to improve the quality of the finite element approximation. We suggest such an adaptive algorithm on the basis of a computable a posteriori error estimate. This estimate is reliable and efficient in the sense that the quotient of the error by the estimate and its inverse are bounded from above. The constants depend on the hardening involved and become larger for decreasing hardening.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050431
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  • 3
    Electronic Resource
    Electronic Resource
    Symmetric coupling of boundary elements and Raviart–Thomas-type mixed finite elements in elastostatics (1996)
    Springer
    Numerische Mathematik 75 (1996), S. 153-174 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N30, 65R20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Both mixed finite element methods and boundary integral methods are important tools in computational mechanics according to a good stress approximation. Recently, even low order mixed methods of Raviart–Thomas-type became available for problems in elasticity. Since either methods are robust for critical Poisson ratios, it appears natural to couple the two methods as proposed in this paper. The symmetric coupling changes the elliptic part of the bilinear form only. Hence the convergence analysis of mixed finite element methods is applicable to the coupled problem as well. Specifically, we couple boundary elements with a family of mixed elements analyzed by Stenberg. The locking-free implementation is performed via Lagrange multipliers, numerical examples are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050235
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  • 4
    Electronic Resource
    Electronic Resource
    Weierstrass formula and zero-finding methods (1995)
    Springer
    Numerische Mathematik 69 (1995), S. 353-372 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65H05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Classical Weierstrass' formula [29] has been often the subject of investigation of many authors. In this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions. We are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros. Conditions for the safe convergence of Weierstrass' method, depending only on initial approximations, are given. In particular, we study polynomials with interval coefficients. Using an interval version of Weierstrass' method enclosures in the form of disks for the complex-valued set containing all zeros of a polynomial with varying coefficients are obtained. We also present Weierstrass-like algorithm for approximating, simultaneously, all zeros of a class of analytic functions in a given closed region. To demonstrate the proposed algorithms, three numerical examples are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050097
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  • 5
    Electronic Resource
    Electronic Resource
    On a class of higher order methods for simultaneous rootfinding of generalized polynomials (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 69-84 
    Details
    ISSN: 0945-3245
    Keywords: 65H05 ; 65H10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Using the argument principle higher order methods for simultaneous computation of all zeros of generalized polynomials (like algebraic, trigonometric and exponential polynomials or exponential sums) are derived. The methods can also be derived following the continuation principle from [3]. Thereby, the unified approach of [7] is enlarged to arbitrary orderN. The local convergence as well as a-priori and a-posteriori error estimates for these methods are treated on a general level. Numerical examples are included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388681
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  • 6
    Electronic Resource
    Electronic Resource
    Numerical approximation of young measuresin non-convex variational problems (2000)
    Springer
    Numerische Mathematik 84 (2000), S. 395-415 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65K10, 65N50, 49M40
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050003
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  • 7
    Electronic Resource
    Electronic Resource
    On quadratic-like convergence of the means for two methods for simultaneous rootfinding of polynomials (1993)
    Springer
    BIT 33 (1993), S. 63-73 
    Details
    ISSN: 1572-9125
    Keywords: AMS(MOS): 65H05 ; CR: 5.15 ; polynomial zeros ; Durand-Kerner method ; simultaneous root finding method ; factorization of polynomials
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Durand-Kerner's method for simultaneous rootfinding of a polynomial is locally second order convergent if all the zeros are simple. If this condition is violated numerical experiences still show linear convergence. For this case of multiple roots, Fraigniaud [4] proves that the means of clustering approximants for a multiple root is a better approximant for the zero and called this Quadratic-Like-Convergence of the Means. This note gives a new proof and a refinement of this property. The proof is based on the related Grau's method for simultaneous factoring of a polynomial. A similar property of some coefficients of the third order method due to Börsch-Supan, Maehly, Ehrlich, Aberth and others is proved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01990344
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  • 8
    Electronic Resource
    Electronic Resource
    Remarks around 50 lines of Matlab: short finite element implementation (1999)
    Springer
    Numerical algorithms 20 (1999), S. 117-137 
    Details
    ISSN: 1572-9265
    Keywords: Matlab program ; 68N15 ; 65N30 ; 65M60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A short Matlab implementation for P 1-x 1 finite elements on triangles and parallelograms is provided for the numerical solution of elliptic problems with mixed boundary conditions on unstructured grids. According to the shortness of the program and the given documentation, any adaptation from simple model examples to more complex problems can easily be performed. Numerical examples prove the flexibility of the Matlab tool.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019155918070
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  • 9
    Electronic Resource
    Electronic Resource
    A reliable modification of the cross rule for rational Hermite interpolation (1992)
    Springer
    Numerical algorithms 3 (1992), S. 29-44 
    Details
    ISSN: 1572-9265
    Keywords: AMS(MOS) 65D05 ; 30E05 ; CR G1.2 ; Rational interpolation ; Newton-Padé approximation ; cross rule ; reliability ; singular rules
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Claessens' cross rule [8] enables simple computation of the values of the rational interpolation table if the table is normal, i.e. if the denominators in the cross rule are non-zero. In the exceptional case of a vanishing denominator a singular block is detected having certain structural properties so that some values are known without further computations. Nevertheless there remain entries which cannot be determined using only the cross rule. In this note we introduce a simple recursive algorithm for computation of the values of neighbours of the singular block. This allows to compute entries in the rational interpolation table along antidiagonals even in the presence of singular blocks. Moreover, in the case of non-square singular blocks, we discuss a facility to monitor the stability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02141913
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    Paper (German National Licenses)
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  • 10
    Electronic Resource
    Electronic Resource
    Domain decomposition for a non-smooth convex minimization problem and its application to plasticity (1997)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Linear Algebra with Applications 4 (1997), S. 177-190 
    Details
    ISSN: 1070-5325
    Keywords: domain decomposition ; convex minimization ; variational inequality ; elastoplasticity ; plasticity with hardening ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Lions's work on the Schwarz alternating method for convex minimization problems is generalized to a certain non-smooth situation where the non-differentiable part of the functionals is additive and independent with respect to the decomposition. Such functionals arise naturally in plasticity where the material law is a variational inequality formulated in L2(Ω). The application to plasticity with hardening is sketched and gives contraction numbers which are independent of the discretization parameter h and of a possible regularization of the non-smooth material law. © 1997 John Wiley & Sons, Ltd.
    Type of Medium: Electronic Resource
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