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  • 1
    Electronic Resource
    Electronic Resource
    A preconditioner for the $h$ - $p$ version of the finite element method in two dimensions (1996)
    Springer
    Numerische Mathematik 75 (1996), S. 59-77 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65F10, 65N30, 65N55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. A preconditioner, based on a two-level mesh and a two-level orthogonalization, is proposed for the $h$ - $p$ version of the finite element method for two dimensional elliptic problems in polygonal domains. Its implementation is in parallel on the subdomain level for the linear or bilinear (nodal) modes, and in parallel on the element level for the high order (side and internal) modes. The condition number of the preconditioned linear system is of order $\max\limits_i(1+\ln {{\textstyle H_ip_i}\over{\textstyle h_i}})^2$ , where $H_i$ is the diameter of the $i$ -th subdomain, $h_i$ and $p_i$ are the diameter of elements and the maximum polynomial degree used in the subdomain. This result reduces to well-known results for the $h$ -version (i.e. $p_i=1$ ) and the $p$ -version (i.e. $h_i=H_i$ ) as the special cases of the $h$ - $p$ version.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050230
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  • 2
    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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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    An additive Schwarz preconditioner for p-version boundary element approximation of the hypersingular operator in three dimensions (2000)
    Springer
    Numerische Mathematik 85 (2000), S. 343-366 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65N55, 65N38, 65F35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. Additive Schwarz preconditioners are developed for the p-version of the boundary element method for the hypersingular integral equation on surfaces in three dimensions. The principal preconditioner consists of decomposing the subspace into local spaces associated with the element interiors supplemented with a wirebasket space associated with the the element interfaces. The wirebasket correction involves inverting a diagonal matrix. If exact solvers are used on the element interiors then theoretical analysis shows that growth of the condition number of the preconditioned system is bounded by $(1+\log p)^2$ for an open surface and $(1+\log p)$ for a closed surface. A modified form of the preconditioner only requires the inversion of a diagonal matrix but results in a further degradation of the condition number by a factor $p(1+\log p)$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110000134
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  • 4
    Electronic Resource
    Electronic Resource
    The h-p version of the finite element method for problems with interfaces (1994)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 37 (1994), S. 1741-1762 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Regularities of the solutions of interface problems in two dimensions are described in the frame of the weighted Sobolev spaces and countably normed spaces. Based upon the regularity of solutions the geometric meshes and the distribution of polynomial degrees are properly designed so that the h-p version of the finite element method for interface problems can lead to the exponential rate of convergence. Numerical results on an elliptic equation with interfaces are presented. The optimal mesh factor, optimal degree factors, and optimal layer factors of the geometric mesh in neighbourhoods of singular points having varied intensities are discussed from both theoretical and practical point of view.
    Additional Material: 12 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620371007
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    Paper (German National Licenses)
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