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  • 1
    Electronic Resource
    Electronic Resource
    The preconditioned GMRES method for systems of coupled FEM-BEM equations (1998)
    Springer
    Advances in computational mathematics 9 (1998), S. 131-144 
    Details
    ISSN: 1572-9044
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We analyze the generalized minimal residual method (GMRES) as a solver for coupled finite element and boundary element equations. To accelerate the convergence of GMRES we apply a hierarchical basis block preconditioner for piecewise linear finite elements and piecewise constant boundary elements. It is shown that the number of iterations which is necessary to reach a given accuracy grows only poly-logarithmically with the number of unknowns.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018921003084
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  • 2
    Electronic Resource
    Electronic Resource
    Wavelet-based preconditioners for boundary integral equations (1998)
    Springer
    Advances in computational mathematics 9 (1998), S. 233-249 
    Details
    ISSN: 1572-9044
    Keywords: boundary integral equations ; prewavelets ; preconditioned conjugate gradient ; additive Schwarz ; hierarchical basis ; 65N55 ; 65N38
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We study simple preconditioners for the conjugate gradient method when used to solve matrix systems arising from some hypersingular and weakly singular integral equations. The preconditioners, which are of the type of hierarchical basis preconditioners, are based on the decomposition of the piecewise-linear (respectively piecewise-constant) functions as the sum of prewavelets (respectively derivatives of prewavelets). We prove that with these preconditioners the preconditioned systems have condition numbers uniformly bounded with respect to the degrees of freedom. Numerical experiments support our analysis.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018993624466
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Boundary integral equations for screen problems in IR3 (1987)
    Springer
    Integral equations and operator theory 10 (1987), S. 236-257 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Here we present a new solution procedure for Helmholtz and Laplacian Neumann screen or Dirichlet screen problems in IR3 via boundary integral equations of the first kind having as unknown the jump of the field or of its normal derivative, respectively, across the screen S. Under the assumption of local finite energy we show the equivalence of the integral equations and the original boundary value problems. Via the Wiener-Hopf method in the halfspace, localization and the calculus of pseudodifferential operators we derive existence, uniqueness and regularity results for the solution of our boundary integral equations together with its explicit behavior near the edge of the screen. We give Galerkin schemes based on our integral equations on S and obtain high convergence rates by using special singular elements besides regular splines as test and trial functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01199079
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  • 4
    Electronic Resource
    Electronic Resource
    An improved boundary element Galerkin method for three-dimensional crack problems (1987)
    Springer
    Integral equations and operator theory 10 (1987), S. 467-504 
    Details
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01201149
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  • 5
    Electronic Resource
    Electronic Resource
    Duality estimates for the numerical solution of integral equations (1989)
    Springer
    Numerische Mathematik 54 (1989), S. 339-353 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65R20 ; CR: G1.9
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We formulate and prove Aubin-Nitsche-type duality estimates for the error of general projection methods. Examples of applications include collocation methods and augmented Galerkin methods for boundary integral equations on plane domains with corners and three-dimensional screen and crack problems. For some of these methods, we obtain higher order error estimates in negative norms in cases where previous formulations of the duality arguments were not applicable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01396766
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  • 6
    Electronic Resource
    Electronic Resource
    Exponential convergence of the hp-version for the boundary element method on open surfaces (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 641-666 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N38, 65R20, 45L10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We analyze the boundary element Galerkin method for weakly singular and hypersingular integral equations of the first kind on open surfaces. We show that the hp-version of the Galerkin method with geometrically refined meshes converges exponentially fast for both integral equations. The proof of this fast convergence is based on the special structure of the solutions of the integral equations which possess specific singularities at the corners and the edges of the surface. We show that these singularities can be efficiently approximated by piecewise tensor products of splines of different degrees on geometrically graded meshes. Numerical experiments supporting these results are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002119900082
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  • 7
    Electronic Resource
    Electronic Resource
    The h-p version of the coupling of finite element and boundary element methods for transmission problems in polyhedral domains (1998)
    Springer
    Numerische Mathematik 80 (1998), S. 87-107 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 45B05, 45E05, 45L10, 65R20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain $\Omega_1$ and with constant coefficients in the exterior domain $\Omega_2 = {\Bbb R}^3 \backslash \overline{\Omega}_1$ . This procedure uses the variational formulation of the differential equation in $\Omega_1$ and involves integral operators on the interface between $\Omega_1$ and $\Omega_2$ . The finite elements are used to obtain approximate solutions of the differential equation in $\Omega_1$ and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050361
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  • 8
    Electronic Resource
    Electronic Resource
    Additive Schwarz algorithms for the p version of the Galerkin boundary element method (2000)
    Springer
    Numerische Mathematik 85 (2000), S. 433-468 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65N55, 65N38
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. We study some additive Schwarz algorithms for the $p$ version Galerkin boundary element method applied to some weakly singular and hypersingular integral equations of the first kind. Both non-overlapping and overlapping methods are considered. We prove that the condition numbers of the additive Schwarz operators grow at most as $(1+\log p)^2$ independently of h, where p is the degree of the polynomials used in the Galerkin boundary element schemes and h is the mesh size. Thus we show that additive Schwarz methods, which were originally designed for finite element discretisation of differential equations, are also efficient preconditioners for some boundary integral operators, which are non-local operators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110000125
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    Paper (German National Licenses)
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  • 9
    Electronic Resource
    Electronic Resource
    Adaptive Coupling and Fast Solution of FEM-BEM Equations for Parabolic-Elliptic Interface Problems (1997)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 20 (1997), S. 403-423 
    Details
    ISSN: 0170-4214
    Keywords: Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we prove an a posteriori error estimate for the symmetric coupling of finite elements and boundary elements applied to linear parabolic-elliptic interface problems. The discontinuous Galerkin method is used for the discretization in time. We present an adaptive algorithm for choosing the mesh size in space and time and we analyse the Hybrid Modified Conjugate Residual (HMCR) method as a solution method for the linear systems which arise. Computational results show that the number of HMCR-iterations grows slowly with the problem size. © 1997 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.
    Additional Material: 4 Ill.
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
  • 10
    Electronic Resource
    Electronic Resource
    Editorial: G. C. Hsiao on his 60th birthday (1997)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 20 (1997), S. 383-384 
    Details
    ISSN: 0170-4214
    Keywords: Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: No Abstract
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
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