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1.
In this paper we consider a singularly perturbed quasilinear boundary value problem depending on a parameter. The problem is discretized using a hybrid difference scheme on Shishkin-type meshes. We show that the scheme is second-order convergent, in the discrete maximum norm, independent of singular perturbation parameter. Numerical experiments support these theoretical results. 相似文献
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In this note a variant of the classical perturbation theorem for singular values is given. The bounds explain why perturbations will tend to increase rather than decrease singular values of the same order of magnitude as the perturbation. 相似文献
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So far there has been no analysis of multigrid methods applied to singularly perturbed Dirichlet boundary-value problems. Only for periodic boundary conditions does the Fourier transformation (mode analysis) apply, and it is not obvious that the convergence results carry over to the Dirichlet case, since the eigenfunctions are quite different in the two cases. In this paper we prove a close relationship between multigrid convergence for the easily analysable case of periodic conditions and the convergence for the Dirichlet case. 相似文献
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A new kind of numerical method based on rational spectral collocation with the sinh transformation is presented for solving parameterized singularly perturbed two-point boundary value problems with one boundary layer. By means of the sinh transformation, the original Chebyshev points are mapped onto the transformed ones clustered near the singular points of the problem. The results from asymptotic analysis as regards the singularity of the solution are employed to determine the parameters in the transformation. Numerical experiments including several nonlinear cases illustrate the high accuracy and efficiency of our method. 相似文献
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I-Dee Chang 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1963,14(2):134-147
Zusammenfassung Die Strömung einer zähen, inkompressiblen und elektrisch leitenden Flüssigkeit über einen rotationssymmetrischen Körper wird studiert mit Hilfe einer singulären Methode der Störungsrechnung. Eine asymptotische, im ganzen Strömungsfeld gültige Lösung wird gegeben für grosse Hartmann-ZahlenM.Die Resultate ergeben folgendes Strömungsbild: Zwei Totwasser-Bereiche von der LängeO (M) und der BreiteO (1) werden vor und nach dem Körper geformt. Sie sind begrenzt durch eine zylindrische Schubschicht, die vom grössten Durchmesser des Körpers aus parabolisch stromaufwärts und stromabwärts anwächst. In einer Entfernung der GrössenordnungO (M) geht diese Schubschicht in eine Wirbelstrasse über, die sich parabolisch ins Unendliche erstreckt. Die Einzelheiten des Strömungsbildes werden analytisch aufgezeigt. Die Wirbelstrasse wird mit derjenigen der klassischen Navier-Stokes-Theorie verglichen. 相似文献
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We analyse the asymptotic behaviour of the solution of a 3Dsingularly perturbed convection–diffusion problem withdiscontinuous Dirichlet boundary data defined in a cuboid. Wewrite the solution in terms of a double series and we obtainan asymptotic approximation of the solution when the singularparameter 0. This approximation is given in terms of a finitecombination of products of error functions and characterizesthe effect of the discontinuities on the small -behaviour ofthe solution in the singular layers. 相似文献
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Consider the equation −ε2Δuε + q(x)uε = f(uε) in , u(∞) < ∞, ε = const > 0. Under what assumptions on q(x) and f(u) can one prove that the solution uε exists and limε→0uε = u(x), where u(x) solves the limiting problem q(x)u = f(u)? These are the questions discussed in the paper. 相似文献
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Shirley Pomeranz Gilbert Lewis Christian Constanda 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,79(1):890-907
An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given. 相似文献
12.
Shirley Pomeranz Gilbert Lewis Christian Constanda 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2005,56(5):890-907
An iterative domain decomposition method is developed to solve a singular perturbation problem. The problem consists of a convection-diffusion equation with a discontinuous (piecewise-constant) diffusion coefficient, and the problem domain is decomposed into two subdomains, on each of which the coefficient is constant. After showing that the boundary value problem is well posed, we indicate a specific numerical implementation of the iterative technique that combines the finite element method on one subdomain with the method of matched asymptotic expansions on the other subdomain. This procedure extends work by Carlenzoli and Quarteroni, which was originally intended for a boundary layer problem with an outer region and an inner region. Our extension carries over to a problem where the domain consists of the outer and inner boundary layer regions plus a region in which the diffusion coefficient is constant and significant in magnitude. An unexpected benefit of our new implementation is its efficiency, which is due to the fact that at each iteration the problem needs to be solved explicitly only on one subdomain. It is only when the final approximation on the entire domain is desired that the matched asymptotic expansions approximation need be computed on the second subdomain. Two-dimensional convergence results and numerical results illustrating the method for a two-dimensional test problem are given.Received: February 12, 2004 相似文献
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Nelly André Itai Shafrir 《Transactions of the American Mathematical Society》2007,359(10):4729-4756
We study the asymptotic behavior, as a small parameter goes to 0, of the minimizers for a variational problem which involves a ``circular-well' potential, i.e., a potential vanishing on a closed smooth curve in . We thus generalize previous results obtained for the special case of the Ginzburg-Landau potential.
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We study the following two phase elliptic singular perturbation problem:
Δuε=βε(uε)+fε,