A fully discrete Galerkin method for Abel-type integral equations
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A fully discrete Galerkin method for Abel-type integral equations Urs V¨ogeli1 · Khadijeh Nedaiasl2 · Stefan A. Sauter1
Received: 26 June 2017 / Accepted: 28 February 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in fractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation. Keywords Abel’s integral equation · Galerkin method · Tensor-Gauss quadrature Mathematics Subject Classification (2000) 45E10 · 65R20 · 65D32
Communicated by: Leslie Greengard Stefan A. Sauter
[email protected] Urs V¨ogeli [email protected] Khadijeh Nedaiasl [email protected] 1
Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190, 8057 Z¨urich, Switzerland
2
Institute for Advanced Studies in Basic Sciences, Zanjan, Iran
U. V¨ogeli et al.
1 Introduction A variety of practical physical models, e.g., in thermal tomography, spectroscopy, astrophysics, can be modelled by Abel-type integral equations provided the problem enjoys symmetries which allow to reduce the equation to a one-dimensional equation (cf. [14, 19]). In this paper, we present a fully discrete Galerkin method for the numerical solution of Abel-type integral equations. The theory and discretization of Abel’s integral equation have been investigated by many authors and a variety of methods are proposed which include, e.g., product integration method [6] and approximation by implicit interpolation [4]. In [2] the solution is approximated by means of an adaptive Huber method which is a kind of product integration method. A Nystr¨om-type method which is based on the trapezoidal rule is analyzed for the Abel integral equation in [10]. Recently, the composite trapezoidal method is applied for weakly singular Volterra integral equations of the first kind [17]. Furthermore, the piecewise polynomial discontinuous Galerkin approximation of a first-kind Volterra integral equation of convolution kernel type for a smooth kernel is studied in [5]. The stability and robustness of the collocation method for Abel integral equation have been discussed in [11]. Less research has been devoted to Galerkin discretizations and to the development of a stability and convergence theory in energy spaces which are fractional-order Sobolev spaces. In our paper, we will propose a variational formulation of generalized Abel’s integral equation and prove continuity and coercivity in the natural energy spaces which are, for these applications, fractional order Sobolev spaces. This allows to employ the classical Lax-Milgram th
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