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  • 1
    Electronic Resource
    Electronic Resource
    On numerical cubatures of singular surface integrals in boundary element methods (1992)
    Springer
    Numerische Mathematik 62 (1992), S. 343-369 
    Details
    ISSN: 0945-3245
    Keywords: 65D30 ; 47G30 ; 45P05 ; 58G15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in ℝ3. The class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three-dimensional domains by the boundary integral equation method. The possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point. This rules out weighted quadrature approaches with a fixed singular weight. For weakly singular integrals it is shown that Duffy's triangular coordinates leadalways to a removal of the kernel singularity. Also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero. These are based on the Rabinowitz-Richter estimates in connection with an asymptotic estimate of domains of analyticity in ℂ2. It is further shown that the modified extrapolation approach due to Lyness is in the weakly singular case always applicable. Corresponding error and asymptotic work estimates are presented. For the weakly singular as well as for Cauchy and hypersingular integrals which e.g. arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates. These rules are hierarchically constructed from “finite part” integration formulas in radial and Gaussian formulas in angular direction. Again, we show how the Rabinowitz-Richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396234
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  • 2
    Electronic Resource
    Electronic Resource
    The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 15-38 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65E05 ; 65N45 ; 45L05 ; 45L10 ; 65R99 ; 65N30 ; 65J10 ; 65R20 ; 47G05 ; 45J05 ; 45F15 ; 47B35 ; CR: G1.9
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389873
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  • 3
    Electronic Resource
    Electronic Resource
    Adaptive boundary element methods for strongly elliptic integral equations (1988)
    Springer
    Numerische Mathematik 53 (1988), S. 539-558 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65N45 ; 65R20 ; 65N99 ; 65N30 ; 65D07 ; 45J05 ; 45B05 ; 45E05 ; 45L10 ; R: 61.9
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Integral operators are nonlocal operators. The operators defined in boundary integral equations to elliptic boundary value problems, however, are pseudo-differential operators on the boundary and, therefore, provide additional pseudolocal properties. These allow the successful application of adaptive procedures to some boundary element methods. In this paper we analyze these methods for general strongly elliptic integral equations and obtain a-posteriori error estimates for boundary element solutions. We also apply these methods to nodal collocation with odd degree splines. Some numerical examples show that these adaptive procedures are reliable and effective.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397551
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  • 4
    Electronic Resource
    Electronic Resource
    Numerical evaluation of singular and finite-part integrals on curved surfaces using symbolic manipulation (1992)
    Springer
    Computing 49 (1992), S. 279-301 
    Details
    ISSN: 1436-5057
    Keywords: Primary 65D30 ; 41A55 ; Secondary 65-04 ; 58G15 ; Numerical cubature ; weakly singular ; Cauchy singular and Hadamard finite part integrals ; symbolic manipulation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir untersuchen die numerische Auswertung aller singulärer Oberflächenintegrale, die beim 3-dimensionalen Rand element-Verfahren entstehen, theoretisch und numerisch. Die Verwendung von dreiecksbezogenen oder lokalen Polarkoordinaten nach Duffy zusammen mit Gaußscher Tensorproduktquadratur erweist sich als effizient und zu verläßich sowohl für die h-als auch die p-Randelemente, wenn schwach singuläre Integrale vorliegen. Oberflächenintegrale mit Cauchyschen oder starken Singularitäten führen wir mit Hilfe von Symbolmanipulation auf solche mit schwachen Singularitäten zurück.
    Notes: Abstract The numerical integration of all singular surface integrals arising in 3-d boundary element methods is analyzed theoretically and computationally. For all weakly singular integrals arising in BEM, Duffy's triangular or local polar coordinates in conjunction with tensor product Gaussian quadrature are efficient and reliable for bothh-andp-boundary elements. Cauchy- and hypersingular surface integrals are reduced to weakly singular ones by analytic regularization which is done automatically by symbolic manipulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02238933
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  • 5
    Electronic Resource
    Electronic Resource
    Galerkin collocation for an improved boundary element method for a plane mixed boundary value problem (1984)
    Springer
    Computing 33 (1984), S. 269-296 
    Details
    ISSN: 1436-5057
    Keywords: 65R20 ; 65N30 ; 45L10 ; 35J05 ; Boundary element method ; singular elements ; error estimates including numerical quadrature
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir beschreiben die numerische Durchführung einer Randelement-Approximation für das ebene gemischte Randwertproblem für harmonische Funktionen. Grundlage unseres Galerkin-Verfahrens ist die direkte Randintegralmethode. Die Genauigkeit des Verfahrens wird dadurch verbessert, daß neben quadratischen Splines geeignete Singularitätenfunktionen als zusätzliche Test- und Ansatzfunktionen verwendet werden. Wir analysieren die Konsistenz der verwendeten numerischen Integrationen und erhalten asymptotische Fehlerabschätzungen der voll diskretisierten Näherungsgleichungen. Diese sind von den gleichen optimalen Ordnungen wie die entsprechenden Galerkin-Fehler.
    Notes: Abstract We present the numerical implementation of a boundary element approximation for the plane mixed boundary value problem of the Laplacian. The performed Galerkin procedure is based on the direct boundary integral method. Its accuracy is improved by using appropriate singularity functions as additional test and trial functions besides quadratic splines. We analize the consistency error of the used numerical integrations and present asymptotic error estimates for the fully discretized numerical scheme which are of the same optimal orders as the Galerkin errors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02242273
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  • 6
    Electronic Resource
    Electronic Resource
    A Galerkin collocation method for some integral equations of the first kind (1980)
    Springer
    Computing 25 (1980), S. 89-130 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir entwickeln hier ein modifiziertes Kollokationsverfahren als vollständige numerische Diskretisierung des Galerkin-Verfahrens zur Lösung von Fredholmschen Integralgleichungen erster Art mit logarithmischem Kern als Hauptteil. Diese Methode verknüpft hohe Genauigkeit der Galerkin-Verfahren mit der Schnelligkeit von Kolookationsmethoden. Die asymptotische Fehleranalysis liefert optimale Konvergenzordnung im Sinne finiter Element-Approximation. Das Verfahren gehört zu den verbesserten Randelement-Methoden, die zur Lösung einer großen Klasse ebener Randwertprobleme unter Verwendung finiter Elemente auf der Randkurve verwendet werden können. Die numerischen Experimente bestätigen hohe Genauigkeit bei kurzen Rechenzeiten.
    Notes: Abstract Here we present a certain modified collocation method which is a fully discretized numerical method for the solution of Fredholm integral equations of the first kind with logarithmic kernel as principal part. The scheme combines high accuracy from Galerkin's method with the high speed of collocation methods. The corresponding asymptotic error analysis shows optimal order of convergence in the sense of finite element approximation. The whole method is an improved boundary integral method for a wide class of plane boundary value problems involving finite element approximations on the boundary curve. The numerical experiments reveal both, high speed and high accuracy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02259638
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  • 7
    Electronic Resource
    Electronic Resource
    The Wiener-Hopf method for systems of pseudodifferential equations with an application to crack problems (1995)
    Springer
    Integral equations and operator theory 23 (1995), S. 294-335 
    Details
    ISSN: 1420-8989
    Keywords: 35Q72 ; 35D10 ; 45E10 ; 47B35 ; 47G30 ; 41A60
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this paper is to develop the Wiener-Hopf method for systems of pseudo-differential equations with “non-constant coefficients” and to apply it to the describtion of the asymptotic behaviour of solutions to boundary integral equations for crack problems when a crack occurs in a linear anisotropic elastic medium. The method was suggested in [15] for scalar pseudo-differential equations with “constant coefficients” and applied in [7] to the crack problems in the isotropic case. The existence and a-priori smoothness of solutions for the anisotropic case has been proved in [11, 12], while the isotropic case has been treated earlier in [7, 25, 41, 50]. Our results improve even those for the isotropic case obtained in [7, 50]. Asymptotic estimates for the behaviour of solutions in the anisotropic case have been obtained in [28] by a different method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01198487
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  • 8
    Electronic Resource
    Electronic Resource
    A hypersingular boundary integral method for two-dimensional screen and crack problems (1990)
    Springer
    Archive for rational mechanics and analysis 112 (1990), S. 363-390 
    Details
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02384079
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  • 9
    Electronic Resource
    Electronic Resource
    Entropy boundary and jump conditions in the theory of sedimentation with compression (1998)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 21 (1998), S. 865-882 
    Details
    ISSN: 0170-4214
    Keywords: Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present an initial-boundary value problem of a quasilinear degenerate parabolic equation for the settling and consolidation of a flocculated suspension. The corresponding definition of generalized solutions is formulated. It is based on an entropy integral inequality in the sense of Kružkov. From this definition, jump and entropy conditions that have to be satisfied at discontinuities, and an entropy condition valid on one boundary of the computational domain are derived. The latter implies a set-valued reformulation of the original boundary condition. It is interpreted geometrically and characterized by the solution of an auxiliary hyperbolic Riemann problem. © 1998 B. G. Teubner Stuttgart-John Wiley & Sons, Ltd.
    Additional Material: 3 Ill.
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  • 10
    Electronic Resource
    Electronic Resource
    Entropy Boundary Conditions in the Theory of Sedimentation of Ideal Suspensions (1996)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 19 (1996), S. 679-697 
    Details
    ISSN: 0170-4214
    Keywords: Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study a non-linear generalized initial-boundary value problem of a scalar conservation law which models the sedimentation of an ideal suspension.
    Additional Material: 7 Ill.
    Type of Medium: Electronic Resource
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