1,721,026 research outputs found
Preconditioning techniques for the iterative solution of scattering problems
No full textWe consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems.We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES
Block decomposition techniques in the generation of adaptive grids
No full textWe consider the problem of the generation of adaptive grids, where, for a given domain Ω, we have to compute a partition that depends on a given map f providing the desired local mesh size. We propose a numerical method for the solution of this problem in the case of planar quadrilateral grids. In particular, a block decomposition method is proposed, by which Ω is first decomposed in
several blocks depending on the shape of the domain and on the size map f, and then the whole grid is obtained as the union of the quadrilateral grids on each block. The algorithm concludes with a smoothing step that depends on the size map f. We present some simple numerical examples to test the proposed method
The Solution of Fredholm Integral Equations of Second Kind by Ritz Approach
No full textA new constructive method to solve Fredholm integral equations of second kind is proposed. This method is based on an analytical procedure that recursively generates a sequence of functions uN, N ∈N. Under suitable conditions, this sequence converges to the solution u∗ of the integral equation under consideration. Several practical versions of this procedure can be obtained by considering different approximation schemes. In particular, following the Ritz approach, the various functions arisen from the analytical method are rewritten by considering a suitable orthogonal system. So that, from the analytical method, we obtain an algorithm for the computation of the coefficients of uN with respect to the orthogonal system taken into account
The heat equation as grid smoothing in a local optimization procedure
No full textWe consider a local optimization technique, where starting from a preliminary version of the grid under consideration, we try to improve the part of the grid that really needs this improvement. When this procedure is performed, the processed grids may result irregular, so a smoothing step must be taken into account. We propose a smoothing approach based on an iterative formula resembling the explicit difference schemes for the heat equation. This is a quite general approach, however to fix the ideas it is
described in the context of quadrilateral grid generation and the variational approach is considered as the base method for the solution of planar grid generation. Some numerical experiments are presented to show the efficiency of the proposed method
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