Elliptic Singular Integro-Differential Operators

Agranovich, M. S.

doi:10.1070/rm1965v020n05abeh001190

Cited by 82 publications

(55 citation statements)

References 32 publications

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“…Finally let t G À/3(A) be the H3 projection of z into M3(A), so (3)(4)(5)(6)(7)(8) l|/|l3<ll*ll3<C||t)||3, and, using (3.6), (3.9) ||/ -z||3 < CAA||z||4 < CAa||ü||3.…”

Section: Iax -A2\mentioning

confidence: 99%

On the asymptotic convergence of collocation methods

Arnold¹,

Wendland²

1983

Math. Comp.

184

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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“…Finally let t G À/3(A) be the H3 projection of z into M3(A), so (3)(4)(5)(6)(7)(8) l|/|l3<ll*ll3<C||t)||3, and, using (3.6), (3.9) ||/ -z||3 < CAA||z||4 < CAa||ü||3.…”

Section: Iax -A2\mentioning

confidence: 99%

On the asymptotic convergence of collocation methods

Arnold¹,

Wendland²

1983

Math. Comp.

184

127

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“…t~O Here Aj, C i, L and 2o are sufficiently smooth complex-valued functions [1]. In the case m=0 these equations reduce to singular integral equations with Cauchy kernels which have been solved with Fourier series in [11] and [30].…”

Section: D"u ~1 Dju A~ ~ + J~o A_ J(z)-~fl~jmentioning

confidence: 99%

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

1985

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Summary.Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.

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“…For this equation preliminary convergence results can be found in [1], [2], [5], [14] and [33]. Our strongly elliptic systems with convolutional principal part contain, in addition, systems of integro-differential equations [3] (see [7]) with constant coefficients, certain singular integral equations, in particular, those of plane elasticity [7,Appendix], [24], [25], [26], [34], Fredholm integral equations of the second kind [6], [8], [11], [12], [13], [17], [27], [35], and also the integro-differential operator of Prandtl's wing theory [16], [17], [18], [24], [35].…”

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confidence: 99%

“…where f = zx(t) + iz2(r), A0, C0 are constant p X p matrices, and where s, respectively T, denotes the arclength on T. The principal symbol of this operator is given by the matrix-valued function [3] aU) = nmi[A0 + c0M foriei> i-2.1.1. The Case of Even m. In the case when m is even, a -m/2 is a nonnegative integer and the strong ellipticity condition (2.2) now reads (2.4) (-l)aRer0K±CoK>a-|2, which must hold for both signs and is equivalent to det(A0 + XC0)*0 forallXe [-1,+1].…”

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confidence: 99%

On the asymptotic convergence of collocation methods with spline functions of even degree

Saranen¹,

Wendland²

1985

Math. Comp.

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Abstract. We investigate the collocation of linear one-dimensional strongly elliptic integro-differential or, more generally, pseudo-differential equations on closed curves by even-degree polynomial splines. The equations are collocated at the respective midpoints subject to uniform nodal grids of the even-degree ß-splines. We prove quasioptimal and optimal order asymptotic error estimates in a scale of Sobolev spaces. The results apply, in particular, to boundary element methods used for numerical computations in engineering applications. The equations considered include Fredholm integral equations of the second and the first kind, singular integral equations involving Cauchy kernels, and integro-differential equations having convolutional or constant coefficient principal parts, respectively.The error analysis is based on an equivalence between the collocation and certain variational methods with different degree splines as trial and as test functions. We further need to restrict our operators essentially to pseudo-differential operators having convolutional principal part. This allows an explicit Fourier analysis of our operators as well as of the spline spaces in terms of trigonometric polynomials providing Babuska's stability condition based on strong ellipticity.Our asymptotic error estimates extend partly those obtained by D. N. Arnold and W. L. Wendland from the case of odd-degree splines to the case of even-degree splines.

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Elliptic Singular Integro-Differential Operators

Cited by 82 publications

References 32 publications

On the asymptotic convergence of collocation methods

On the asymptotic convergence of collocation methods

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

On the asymptotic convergence of collocation methods with spline functions of even degree

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