In this paper we consider the (2D and 3D) exterior problem for the non homogeneous wave equation, with a Dirichlet boundary condition and non homogeneous initial conditions. First we derive two alternative boundary integral equation formulations to solve the problem. Then we propose a numerical approach for the computation of the extra "volume" integrals generated by the initial data. To show the efficiency of this approach, we solve some test problems by applying a second order Lubich discrete convolution quadrature for the discretization of the time integral, coupled with a classical collocation boundary element method. Some conclusions are finally drawn.
Abstract. To solve one-dimensional linear weakly singular integral equations on bounded intervals, with input functions which may be smooth or not, we propose to introduce first a simple smoothing change of variable, and then to apply classical numerical methods such as product-integration and collocation based on global polynomial approximations. The advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well-conditioned.
In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.
SUMMARYIn this paper, we propose an efficient strategy to compute nearly singular integrals over planar triangles in R 3 arising in boundary element method collocation. The strategy is based on a proper use of various non-linear transformations, which smooth or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r , 1/r 2 and 1/r 3 , r = x鈭抷 being the distance between a fixed near observation point x and a generic point y of a triangular element. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive.
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