Solution of Boundary Value Problems by Integral Equations of the First Kind

Hsiao, George; MacCamy, R. C.

doi:10.1137/1015093

Cited by 167 publications

(94 citation statements)

References 4 publications

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“…We also estimate the conditioning of the discrete equations. Next we generalize the estimates to the case of certain systems of equations of a form which arises frequently in applications, e.g., in Fichera's method with Fredholm integral equations of the first kind [37], [50], [51], [52], [53]. We then verify that strongly elliptic pseudodifferential operators satisfy all the assumptions made in the analysis.…”

mentioning

confidence: 85%

“…The scalar case with p = q = 1, B = 1, X0 = 1, L = 0 is Symm's integral equation of conformai mapping [40], [45], [48], [51], [95], [100]. In [31] it is shown that the numerical conditioning is superior with B = 1, X0 = 1 rather than B = 0, X0 = 0 and scaling of T. As a system, (2.3.12) is used in viscous flow problems, particularly in connection with Stokes flows [50]- [53], electrostatics [73], [85], [86], acoustics [38], plane elasticity [20], [32], [51], [56], [73], plate bending [51], [52], and torsion problems [56] [50], [51], [53], [54], [55], [73], [85], [86], [100]. However, only preliminary convergence results have appeared for the collocation method [1], [2], [5, p. 271 ff.…”

Section: Iax -A2\mentioning

confidence: 99%

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On the asymptotic convergence of collocation methods

Arnold¹,

Wendland²

1983

Math. Comp.

182

127

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Abstract. We prove quasioptimal and optimal order estimates in various Sobolev norms for the approximation of linear strongly elliptic pseudodifferential equations in one independent variable by the method of nodal collocation by odd degree polynomial splines. The analysis pertains in particular to many of the boundary element methods used for numerical computation in engineering applications. Equations to which the analysis is applied include Fredholm integral equations of the second kind, certain first kind Fredholm equations, singular integral equations involving Cauchy kernels, a variety of integro-differential equations, and two-point boundary value problems for ordinary differential equations. The error analysis is based on an equivalence which we establish between the collocation methods and certain nonstandard Galerkin methods. We compare the collocation method with a standard Galerkin method using splines of the same degree, showing that the Galerkin method is quasioptimal in a Sobolev space of lower index and furnishes optimal order approximation for a range of Sobolev indices containing and extending below that for the collocation method, and so the standard Galerkin method achieves higher rates of convergence.1. Introduction. In this paper we apply the method of nodal collocation by odd degree polynomial splines to systems of strongly elliptic pseudodifferential equations on closed curves and to two-point boundary value problems for ordinary differential equations. (By splines we always mean smoothest splines.) The former class of problems encompasses many of the boundary integral equations discretized by boundary element methods including some first kind and second kind Fredholm integral equations, singular integral equations with Cauchy kernels, and certain integro-differential equations such as the normal derivative of the double layer potential and the operator of Prandtl's wing theory.For spline approximation of such strongly elliptic equations via standard Galerkin procedures or via the corresponding numerically integrated Galerkin-collocation methods the asymptotic error analysis is already rather completely developed; see

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mentioning

confidence: 85%

Section: Iax -A2\mentioning

confidence: 99%

On the asymptotic convergence of collocation methods

Arnold¹,

Wendland²

1983

Math. Comp.

182

127

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“…This is different than the proof available in [4]. The following two lemmas may also be found in [3].…”

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confidence: 79%

“…This is a standard argument. (b) The uniqueness of problems I and II are also standard, as is the existence of 4> and (i//, C) (see for example [3,4]). Our low-frequency theory is based on the existence proofs presented in [3], To present our main result we need several lemmas.…”

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confidence: 99%

Inverse scattering for an exterior Dirichlet problem

Hariharan¹

1982

Quart. Appl. Math.

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Abstract. In this paper we consider scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current. The aim of the paper is to obtain information such as the location and shape of the cylinder with a knowledge of a far-field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft-wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. We present an improved low-frequency estimate for the solution of this problem using integral equation methods, and our calculations on inverse scattering are accurate to this estimate. The far-field measurements are related to the solutions of boundary integral equations in the low-frequency situation. These solutions can be expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations can be related to the far-field coefficients. The first far-field coefficient leads to the calculation of the distance between the source and the cylinder. The other coefficients are determined by placing the source in a different location and using the corresponding new far-field measurements.Introduction. In this note we are interested in the scattering due to a metallic cylinder which is in a field caused by a wire carrying a periodic current. The goal is to obtain

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“…In order to avoid difficulties of nonuniqueness when the interior domain has conformai radius one, we follow [6], [11], [14] in seeking to express « as a normalized single layer potential plus a constant. That is, we seek a function xp on L" of mean value zero and a real number co such that (1.2) u(z) = -^-flog\z-y\xp(y)dsv + oo, zER2.…”

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confidence: 99%

A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method

Arnold¹

1983

Math. Comp.

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Abstract. We consider a Galerkin method for functional equations in one space variable which uses periodic cardinal splines as trial functions and trigonometric polynomials as test functions.We analyze the method applied to the integral equation of the first kind arising from a single layer potential formulation of the Dirichlet problem in the interior or exterior of an analytic plane curve. In constrast to ordinary spline Galerkin methods, we show that the method is stable, and so provides quasioptimal approximation, in a large family of Hubert spaces including all the Sobolev spaces of negative order. As a consequence we prove that the approximate solution to the Dirichlet problem and all its derivatives converge pointwise with exponential rate.

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Solution of Boundary Value Problems by Integral Equations of the First Kind

Cited by 167 publications

References 4 publications

On the asymptotic convergence of collocation methods

On the asymptotic convergence of collocation methods

Inverse scattering for an exterior Dirichlet problem

A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method

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