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1.
Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation
of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset
of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions
with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we
obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin
method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of
singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not
even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular)
meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points
in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory.
Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000 相似文献
2.
Summary. In this paper we describe and analyse a class of spectral methods, based on spherical polynomial approximation, for second-kind
weakly singular boundary integral equations arising from the Helmholtz equation on smooth closed 3D surfaces diffeomorphic
to the sphere. Our methods are fully discrete Galerkin methods, based on the application of special quadrature rules for computing
the outer and inner integrals arising in the Galerkin matrix entries. For the outer integrals we use, for example, product-Gauss
rules. For the inner integrals, a variant of the classical product integration procedure is employed to remove the singularity
arising in the kernel. The key to the analysis is a recent result of Sloan and Womersley on the norm of discrete orthogonal
projection operators on the sphere. We prove that our methods are stable for continuous data and superalgebraically convergent
for smooth data. Our theory includes as a special case a method closely related to one of those proposed by Wienert (1990)
for the fast computation of direct and inverse acoustic scattering in 3D.
Received May 29, 2000 / Revised version received March 26, 2001/ Published online December 18, 2001 相似文献
3.
Pouria Assari 《Applicable analysis》2013,92(11):2064-2084
The present work proposes a numerical method to obtain an approximate solution of non-linear weakly singular Fredholm integral equations. The discrete Galerkin method in addition to thin-plate splines established on scattered points is utilized to estimate the solution of these integral equations. The thin-plate splines can be regarded as a type of free shape parameter radial basis functions which create an efficient and stable technique to approximate a function. The discrete Galerkin method for the approximate solution of integral equations results from the numerical integration of all integrals in the method. We utilize a special accurate quadrature formula via the non-uniform composite Gauss-Legendre integration rule and employ it to compute the singular integrals appeared in the scheme. Since the approach does not need any background meshes, it can be identified as a meshless method. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates. 相似文献
4.
Midpoint collocation for Cauchy singular integral equations 总被引:1,自引:0,他引:1
G. A. Chandler 《Numerische Mathematik》1992,62(1):483-509
Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems 相似文献
5.
A meshless local Galerkin method for the numerical solution of Hammerstein integral equations based on the moving least squares technique
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Pouria Assari 《Journal of Applied Analysis & Computation》2019,9(1):75-104
In this paper, a computational scheme is proposed to estimate the solution of one- and two-dimensional Fredholm-Hammerstein integral equations of the second kind. The method approximates the solution using the discrete Galerkin method based on the moving least squares (MLS) approach as a locally weighted least squares polynomial fitting. The discrete Galerkin technique for integral equations results from the numerical integration of all integrals in the system corresponding to the Galerkin method. Since the proposed method is constructed on a set of scattered points, it does not require any background meshes and so we can call it as the meshless local discrete Galerkin method. The implication of the scheme for solving two-dimensional integral equations is independent of the geometry of the domain. The new method is simple, efficient and more flexible for most classes of nonlinear integral equations. The error analysis of the method is provided. The convergence accuracy of the new technique is tested over several Hammerstein integral equations and obtained results confirm the theoretical error estimates. 相似文献
6.
Summary In this paper we consider the so-called generalized airfoil singular integral equation, with a smooth input function, and present ad hoc quadrature rules to compute efficiently the elements of collocation and Galerkin matrices.Work sponsored by the Italian Ministry of Education 相似文献
7.
A hypersingular boundary integral equation of the first kind on an open surface piece Γ is solved approximately using the Galerkin method. As boundary elements on rectangles we use continuous, piecewise bilinear functions which vanish on the boundary of Γ. We show how to compensate for the effect of the edge and corner singularities of the true solution of the integral equation by using an appropriately graded mesh and obtain the same convergence rate as for the case of a smooth solution. We also derive asymptotic error estimates in lower-order Sobolev norms via the Aubin–Nitsche trick. Numerical experiments for the Galerkin method with piecewise linear functions on triangles demonstrate the effect of graded meshes and show experimental rates of convergence which underline the theoretical results. 相似文献
8.
In this paper we analyze the discretization of optimal control problems governed by convection-diffusion equations which are
subject to pointwise control constraints. We present a stabilization scheme which leads to improved approximate solutions
even on corse meshes in the convection dominated case. Moreover, the in general different approaches “optimize-then- discretize”
and “discretize-then-optimize” coincide for the proposed discretization scheme. This allows for a symmetric optimality system
at the discrete level and optimal order of convergence. 相似文献
9.
When n>2 it is well known that the spherical partial sums of n-fold Fourier integrals of a characteristic function of the ball D={x:|x|2<1} do not converge at the origin. In the mathematical literature this result is called “the Pinsky phenomenon”. In 1993 Pinsky
established necessary and sufficient conditions for a piecewise smooth function, supported on D, which guarantee the convergence at the origin its spherical partial sums. We prove this result for nonspherical partial
sums, i.e. for Fourier integrals under summation over domains bounded by level surfaces of elliptic polynomials. 相似文献
10.
In the framework of the Jacobi-weighted Besov spaces, we analyze the lower and upper bounds of errors in the h–p version of boundary element solutions on quasiuniform meshes for elliptic problems on polygons. Both lower bound and upper
bound are optimal in h and p, and they are of the same order. The optimal convergence of the h–p version of boundary element method with quasiuniform meshes is proved, which includes the optimal rates for h version with quasiuniform meshes and the p version with quasiuniform degrees as two special cases.
Dedicated to Professor Charles Micchelli on the occasion of his sixtieth birthday
Mathematics subject classification (2000) 65N38.
Benqi Guo: The work of this author was supported by NSERC of Canada under Grant OGP0046726 and was complete during visiting
Newton Institute for Mathematical Sciences, Cambridge University for participating in special program “Computational Challenges
in PDEs” in 2003.
Norbert Heuer: This author is supported by Fondecyt project No. 1010220 and by the FONDAP Program (Chile) on Numerical Analysis.
Current address: Mathematical Sciences, Brunel University, Uxbridge, U.K. 相似文献
11.
Summary We present and analyze methods for the accurate and efficient evaluation of weakly, Cauchy and hypersingular integrals over piecewise analytic curved surfaces in 3.The class of admissible integrands includes all kernels arising in the numerical solution of elliptic boundary value problems in three-dimensional domains by the boundary integral equation method. The possibly not absolutely integrable kernels of boundary integral operators in local coordinates are pseudohomogeneous with analytic characteristics depending on the local geometry of the surface at the source point. This rules out weighted quadrature approaches with a fixed singular weight.For weakly singular integrals it is shown that Duffy's triangular coordinates leadalways to a removal of the kernel singularity. Also asymptotic estimates of the integration error are provided as the size of the boundary element patch tends to zero. These are based on the Rabinowitz-Richter estimates in connection with an asymptotic estimate of domains of analyticity in 2.It is further shown that the modified extrapolation approach due to Lyness is in the weakly singular case always applicable. Corresponding error and asymptotic work estimates are presented.For the weakly singular as well as for Cauchy and hypersingular integrals which e.g. arise in the study of crack problems we analyze a family of product integration rules in local polar coordinates. These rules are hierarchically constructed from finite part integration formulas in radial and Gaussian formulas in angular direction. Again, we show how the Rabinowitz-Richter estimates can be applied providing asymptotic error estimates in terms of orders of the boundary element size.Partially supported by the Priority Research Programme Boundary Element Methods of the German Research Foundation DFG under Grant No. We 659/16-1 (guest programme) and under AFOSR-grant 89-0252. 相似文献
12.
13.
This paper deals with a semi-discrete version of the Galerkin method for the single-layer equation in a plane, in which the
outer integral is approximated by a quadrature rule. A feature of the analysis is that it does not require high precision
quadrature or exceptional smoothness of the data. Instead, the assumptions on the quadrature rule are that constant functions
are integrated exactly, that the rule is based on sufficiently many quadrature points to resolve the approximation space,
and that the Peano constant of the rule is sufficiently small. It is then shown that the semi-discrete Galerkin approximation is well posed. For the
trial and test spaces we consider quite general piecewise polynomials on quasi-uniform meshes, ranging from discontinuous
piecewise polynomials to smoothest splines. Since it is not assumed that the quadrature rule integrates products of basis
functions exactly, one might expect degradation in the rate of convergence. To the contrary, it is shown that the semi-discrete
Galerkin approximation will converge at the same rate as the corresponding Galerkin approximation in the and norms.
Received March 15, 1996 / Revised version received June 2, 1997 相似文献
14.
Summary. We analyze the boundary element Galerkin method for weakly singular and hypersingular integral equations of the first kind
on open surfaces. We show that the hp-version of the Galerkin method with geometrically refined meshes converges exponentially
fast for both integral equations. The proof of this fast convergence is based on the special structure of the solutions of
the integral equations which possess specific singularities at the corners and the edges of the surface. We show that these
singularities can be efficiently approximated by piecewise tensor products of splines of different degrees on geometrically
graded meshes. Numerical experiments supporting these results are presented.
Received December 19, 1996 / Revised version received September 24, 1997 / Published online August 19, 1999 相似文献
15.
V. J. Ervin E. P. Stephan S. Abou El-Seoud 《Mathematical Methods in the Applied Sciences》1990,13(4):291-303
A weakly singular integral equation of the first kind on a plane surface piece Γ is solved approximately via the Galerkin method. The determination of the solution of this integral equation (with the single-layer potential) is a classical problem in physics, since its solution represents the charge density of a thin, electrified plate Γ loaded with some given potential. Using piecewise constant or piecewise bilinear boundary elements we derive asymptotic estimates for the Galerkin error in the energy norm and analyse the effect of graded meshes. Estimates in lower order Sobolev norms are obtained via the Aubin–Nitsche trick. We describe in detail the numerical implementation of the Galerkin method with both piecewise-constant and piecewise-linear boundary elements. Numerical experiments show experimental rates of convergence that confirm our theoretical, asymptotic results. 相似文献
16.
Summary This article analizes the convergence of the Galerkin method with polynomial splines on arbitrary meshes for systems of singular integral equations with piecewise continuous coefficients inL
2 on closed or open Ljapunov curves. It is proved that this method converges if and, for scalar equations and equidistant partitions, only if the integral operator is strongly elliptic (in some generalized sense). Using the complete asymptotics of the solution, we provide error estimates for equidistant and for special nonuni-form partitions. 相似文献
17.
In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non‐singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and convergent scheme in which there are non‐singular integrals. We have also presented another point of view under which the method can be considered as a special quadrature formula method for the standard piecewise linear Galerkin approximation of the weakly singular single‐layer potential. In the present paper, we extend the results given in the previous paper to more general cases for which the Laplace problem is set on any ??∞ domains. We prove that the properties of stability and convergence remain valid. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
18.
19.
Johannes Elschner 《Mathematische Nachrichten》1988,139(1):309-319
This paper analyses the convergence of spline approximation methods for strongly elliptic singular integral equations on a finite interval. We consider collocation by smooth polynomial splines of odd degree multiplied by a weight function and a Galerkin-Petrov method with spline trial functions of even degree and piecewise constant test functions. We prove the stability of the methods in weighted Sobolev spaces and obtain the optimal orders of convergence in the case of graded meshes. 相似文献
20.
Nicolas Th. Varopoulos 《Milan Journal of Mathematics》2008,76(1):419-429
In this note I give an extension of the T1-Theorem of David and Journé. In this extension the standard estimates on the kernel
are replaced by “average standard estimates” but the classical proof is essentially identical. I then use that extension to
prove the Lp-boundedness of some natural rough singular integrals. The applications of these singular integrals in Potential Theory has
already been described in my previous paper in the area.
Received: September 2008 相似文献