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  • 1
    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
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  • 2
    ISSN: 1432-0673
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
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  • 3
    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
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  • 4
    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
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  • 5
    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
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  • 6
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Although the plane boundary value problem for the Laplacian with given Dirichlet data on one part Γ2 and given Neumann data on the remaining part Γ2 of the boundary is the simplest case of mixed boundary value problems, we present several applications in classical mathematical physics. Using Green's formula the problem is converted into a system of Fredholm integral equations for the yet unknown values of the solution u on Γ2 and the also desired values of the normal derivatie on Γ1. One of these equations has principal part of the second kind, whereas that one of the other is of the first kind. Since any improvement of constructive methods requires higher regularity of u but, on the other hand, grad u possesses singularities at the collision points Γ1 ∩ Γ2 even for C∞ data, u is decomposed into special singular terms and a regular rest. This is incorporated into the integral equations and the modified system is solved in appropriate Sobolev spaces. The solution of the system requires to solve a Fredholm equation of the first kind on the arc Γ2 providing an improvement of regularity for the smooth part of u. Since the integral equations form a strongly elliptic system of pseudodifferential operators, the Galerkin procedure converges. Using regular finite element functions on Γ1 and Γ2 augmented by the special singular functions we obtain optimal order of asymptotic convergence in the norm corresponding to the energy norm of u and also superconvergence as well as high orders in smoother norms if the given data are smooth (and not the solution).
    Additional Material: 8 Ill.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    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.
    Type of Medium: Electronic Resource
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  • 8
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L2. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Chichester, West Sussex : Wiley-Blackwell
    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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