Über die Näherungsweise Lösung von Linearen Funktionalgleichungen

Bruhn, Gerhard; Wendland, Wolfgang L.

doi:10.1007/978-3-0348-5821-2_16

Cited by 20 publications

(11 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since (3.11) has this particular structure, corresponding boundary element approximations with piecewise linear continuous splines μ h ∈ S h on a quasi regular family of grids where h denotes the maximal meshwidth, and point collocation at the grid points converges to μ in C 0 (Γ) if h tends to zero [4]. Also piecewise constant approximation and point collocation at the element midpoints converges (see the arguments in [27] for the two-dimensional case).…”

Section: Then the Fredholm Radius Is Infinitementioning

confidence: 95%

On the Double Layer Potential

Wendland

2009

Analysis, Partial Differential Equations and Applications

View full text Add to dashboard Cite

Abstract. Neumann's classical integral equation with the double layer potential operator is considered on different spaces of boundary charges, such as continuous data, L 2 (Γ) and energy trace spaces. Corresponding known results for different classes of boundaries are collected and discussed in view of their consequences for collocation or Galerkin boundary element methods. Mathematics Subject Classification (2000). 31A10, 45B05, 45A05.

show abstract

Section: Then the Fredholm Radius Is Infinitementioning

confidence: 95%

On the Double Layer Potential

Wendland

2009

Analysis, Partial Differential Equations and Applications

View full text Add to dashboard Cite

show abstract

“…They describe the acoustic motion of a non-viscous fluid initially at rest, and excited by a potential force, that is : 0 and f In that case, an equation governing the acoustic pressure II is easily obtained ( 17) If another quantity ~ , the velocity potential, is introduced All this course is devoted to this simple forms of acoustic equations. Nevertheless, the main ideas of the mathematical analysis presented hers enable to get analytical, and even numerical results for more complex acoustical motions.…”

Section: -5 the D'alembert Equation And The Helmholtz Equati~mentioning

confidence: 99%

“…To solve (17) is the Bessel function of order (n-2)/2. The integration can now be performed by the residue method, using the contour {-R <;;; ~ <;;; + R ; ~ = Rei 9 , 0 <;;; 8 <;;; TI ; R -+ oo } and we are left with :…”

Section: 1 -Elementary Solutions and Elementary Kernels Of Thmentioning

confidence: 99%

See 1 more Smart Citation

Theoretical Acoustics and Numerical Techniques

Filippi¹

1983

View full text Add to dashboard Cite

This work is suhject to copyright.AII rights are reserved, whether the whole or part of the material is concemed specifically those of translation, reprinting, re-use of illustrations, hroadcasting, reproduction hy photocopying machine or similar means, and storage in data hanks. Boundary Value Problems Analysis and Pseudo-Differential Operators in Acoustics 1/ ACOUSTICS AND CLASSICAL MATHEMATICSLet, first, have a brief survey of the mathematical problems appearing in Acoustics. The time-dependant governing equation is of hyerbolice type (a much less simple case than the parabolic type ), and unbounded domains must be considered as soon as environmental acoustics or under-water propagation are concerned.Because of the difficulty to solve the wave equation, and because lots of noise and sound sources are periodic (or can be considered as periodic ), the Helmholtz equation is more frequently used. If the propagation domain is bounded, resonnances appear. If the propagation domain is unbounded, the total energy involved is unbounded, too. For these reasons, the use of the classical variational techniques is much less easy than for the heat equation, static solid mechanics, or incompressible fluid dynamics. Another difficulty is that as soon as energy is lost within the boundaries -and this generally the case -the operators involved are not self-adjoint : consequently, the powerful spectral theory does not apply in its classical form.Another type of difficulty will appear if acoustic energy can Another classical method is to use asymptotic expansions with respect to the distance or the frequency, or other characteristic parame- ters. The Geometrical Theory of Diffraction belongs to this category ;~oughthey are based on considerations which seem satisfactory to the physicist, the results are not always proved.In the last fifteen years, boundary integral equations have been used in acoustic diffraction [and, simultaneously, in electromagnetism ). More or less simple numerical procedures have been adopted, but their convergence has been proved recently, only. At the same period, finite elementsmethods have been succesfully used for solving problems in bounded domains ; for unbounded domains, these methods appear to be less efficient than the boundary integral equations method. 2/ THE MODERN MATHEMATICAL ANALYSISAs far as the data [boundary surfaces, source distribution, space characteristic parameters of the physical medium, ... )are described by sufficiently regular functions, and if local boundary conditions are considered, the existence and uniqueness theorems of the solution can be proved, using very classical mathematics. But, in many practical cases, the necessary· regularity hypothesis are not fulfilled : the boundaries can have corners ; the actual sources are efficiently described by distribuPreface v tions (think of multipole sources encountered in jet noise description J.Non local boundary conditions are of practical interest : this is the case of domains bounded by vibrating structures ; another example is...

show abstract

“…The principal symbol is the identity matrix, so the equations are trivially strongly elliptic and our results apply. However, they can easily be obtained in this case from well-known results in [6], [28] in combination with approximation estimates (2.1.4).…”

Section: Iax -A2\mentioning

confidence: 99%

On the Asymptotic Convergence of Collocation Methods

Arnold¹,

Wendland²

1983

Mathematics of Computation

View full text Add to dashboard Cite

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

show abstract

Über die Näherungsweise Lösung von Linearen Funktionalgleichungen

Cited by 20 publications

References 7 publications

On the Double Layer Potential

On the Double Layer Potential

Theoretical Acoustics and Numerical Techniques

On the Asymptotic Convergence of Collocation Methods

Contact Info

Product

Resources

About