K. E. Atkinson, "A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions," in Numerical Solution of Integral Equations, M. Goldberg, Ed. New York: Plenum, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... fN (v j;K ) j;K (q) For the collocation method, fN can be shown to satisfy when S is a smooth surface; this is based on results from Nedelec[8] For piecewise smooth surfaces, it has been shown to be at least O( N ) see Atkinson[3] But for the Nystrom method (see Atkinson[5]) fN k1 Ck(K Gamma KN )fk1 : 12) Thus for the collocation method, we have the alternative error bound fN (v i ) j Ck(K Gamma KN )fk1 : With this as motivation, we examine the error k(K Gamma KN )fk1 . Atkinson[3] has shown that k j D s mK Theta D t mK j Gamma j D s f mK Theta D t f ....

Atkinson, K., A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, edited by M. Golberg, Plenum Press, New York, 1990.

....time [60] In fact, if a BEM equation does not converge to a suitable accuracy, expansion basis should be updated and a new MoM matrix should be assembled. Denseness of this matrix as well as non compactness of the kernel renders iterative matrix solvers to be unreasonably expensive to use [61]. A great deal of research has been done in accelerating BEM techniques. Examples of such fast algorithms are: applications of fast n body problem algorithms including multipole accelerated algorithm [62] and Appel s hierarchical algorithm [65] These algorithms have O(N) cost per iteration for ....

....us define inner product (section 2.1.2) as: 4.19) For the capacitance extraction problem, linear operator in equation (2.27) takes the form integral operator on an interval: 4.20) where G is defined as in (4.15) For integral operators in general, matrix in equation (2. 33) is a full matrix [61], with the cost of assembling such matrix itself being of O(N ) and cost of solving equation being of O(N ) This cost can be substantially reduced by choosing orthogonal wavelets with compact support as basis functions and successive thresholding of stiffness matrix that results in sparse ....

K.E. Atkinson, "A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions," in Numerical Solution of Integral Equations, M. Goldberg, ed., Plenum Press, 1990.

....E#cient iterative solvers such as the conjugate gradient and multigrid methods have only been shown to be applicable to the case where the underlying integral operator is compact. In the case of K in (1. 2) being compact, successful applications of iterative methods have been reported; see [5, 2, 21, 25, 30, 32]. # Received January 12, 1994. Accepted for publication June 15, 1994. Communicated by L. Reichel. Department of Statistics and Computational Mathematics, Victoria Building, The University of Liverpool, Brownlow Hill, Liverpool L69 3BX, England. Email : chen scmp.scm.liv.ac.uk) 76 ETNA ....

K. E. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, M. Goldberg, ed., Plenum Press, 1990.

....65R20, 65N35, 45E05. 1 Introduction Let us motivate the following discussion by considering a well known integral equation related to Poisson s Equation: 1.1) #u = f in D, u = g on B, where D # R 3 is a bounded open nonempty domain with su#ciently regular boundary B = # D. Following [6], we utilize the fundamental solution s(x, y) 1 4# 1 x y to obtain a particular solution (which does not in general satisfy the above boundary condition) via the Poisson integral formula (1.2) v(x) # D s(x, y) f(y) dy, 1 Partially supported by the National Science Foundation ....

Atkinson, K. E. (1990): A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions. In: Numerical Solution of Integral Equations. M. Goldberg editor, Plenum Press, New York.

....solution of the following exterior problem in R 3 with Neumann boundary conditions: #u =0inD, #u ## = g on B, where D is the region exterior to the unit sphere centered at the origin, B = #D and #(y) indicates the outer normal of #D at y # #D. It it well known, see, e.g. Atkinson [2, 3] that u satisfies the following Fredholm integral equation of the second kind: 2#u(x) Z B K(x, y)u(y) dy) Z B g(y) 1 x y (dy) where is the standard surface measure (surface element) of integration and K(x, y) # ##(y) 1 x y is the kernel of the so called double layer ....

K. E. Atkinson. A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions. In M. Goldberg, editor, Numerical Solution of Integral Equations, pages 1--34, New York, 1990. Plenum Press.

....65D30, 65R20, 65N35, 45E05 1 Introduction Let us motivate the following discussion by considering a well known integral equation related to Poisson s Equation: #u = f in D, u = g on B, where D # R 3 is a bounded open nonempty domain with su#ciently regular boundary B = # D. Following [5], we utilize the fundamental solution s(x, y) 1 4# 1 x y to obtain a particular solution (which does not in general satisfy the above boundary condition) via the Poisson integral formula v(x) # D s(x, y) f(y) dy, 1) 1 Partially supported by the National Science Foundation ....

K. E. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, M. Goldberg, ed., New York, 1990, Plenum Press, pp. 1--34.

....the discussion, let us return to the potential operators described in Example 5. A typical integral equation of the second kind is obtained by setting L : 1 1 2# K 2 and considering (1) over F = C(S) This is a boundary integral equation formulation of Poisson s equation, see, e.g. [5]. A typical discretization involves a triangulation of S and a finite element type approach with respect to this triangulation. Hence, we consider a set of basis functions # j j#J #Fand a set of collocation points s i i#I #Sof equal cardinality. Typically, the basis functions ....

K. E. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, M. Goldberg, ed., New York, 1990, Plenum Press.

....equations are mostly coming from two distinct communities. Nevertheless, the techniques these communities are using come from the same basic framework. There are a variety of ways to enter the enormous literature on boundary integral equations. There are the excellent recent surveys by Atkinson [2] and Sloan [33] I have used Chapter 10 of [25] in a graduate course for the numerical solutions of PDE. A simple exposition with example can be found on page 321 327 of [1] The chart below indicates the major two communities involved with large boundary integral equation applications Community ....

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solutions of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990.

....equations are mostly coming from two distinct communities. Nevertheless, the techniques these communities are using come from the same basic framework. There are a variety of ways to enter the enormous literature on boundary integral equations. There are the excellent recent surveys by Atkinson [2] and Sloan [34] I have used Chapter 10 of ( 25] in a graduate course for the numerical solutions of PDE. A simple exposition with example can be found on pages 321 327 of ( 1] Other examples can be found in Bendali, 1984; Brebbia, 1984; Canning, 1990a, 1990b 1993; Harrington, 1990; Hess, ....

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solutions of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990.

....transonic lifting potential flow. He used GMRES with preconditioner. He reported good accuracy after ten iterations. Ken Atkinson at the University of Iowa has been developing methods for the solutions of boundary integral equations, though the matrices he has solved were not huge (under 10,000) [1] B. Alpert at Lawrence Berkeley Laboratories, Per Lotstdedt at the Aircraft Division of SAAB Scania in Sweden, and Hans Munthe Kaas who was working for the Norwegian Hydrodynamic laboratories in Trondheim (now at the University of Bergen, Norway) were all working on panel methods (or something ....

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solutions of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990.

....2 operators are involved in the analysis. The compactness leads to a fairly well developed mathematical analysis, for example, see Gunter (1967) Mikhlin (1970) and Pogorzelski (1966) A wide variety of numerical methods have been used to study three dimensional BIE problems (e.g. see Atkinson [11]) and most numerical methods for the problems can be considered to be of collocation or Galerkin type. The most general convergence results for collocation methods are given by Wendland (1985) He used polynomials of degree k to approximate surfaces and polynomials of degree d to approximate ....

K. Atkinson(1990) A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Press, 1--34.

....reliable convergence of the underlying iterative method. To aid in insuring rapid iteration convergence, multilevel and local inversion preconditioners have been developed for discretized integral equations, and these techniques substantially accelerate convergence rates for fine discretizations [3, 4]. However, multipole accelerated iterative methods have provided the greatest benefit for engineering applications, which, because of the low accuracy requirements, rarely use fine discretizations. In this paper, we address the problem of preconditioning systems of equations generated from coarse ....

K. E. Atkinson, "A survey of boundary integral equation methods for the numerical solution of laplace's equation in three dimensions," in Numerical Solution of Integral Equations (M. A. Goldberg, ed.), pp. 1--34, New York: Plenum Press, 1990.

....methods. The estimates are surprisingly difficult since O(h 3 ) terms have to be shown to cancel; this does not occur in the expansion of the standard rules. 1. Introduction Efficient numerical approximations of surface integrals are important in boundary element methods, see, e.g. Atkinson [1, 2], Georg and Widmann [4] Hackbusch [5] Recently, Georg [3] introduced a new approach to the numerical quadrature of surface integrals. It was assumed that the surface B ae R 3 was modeled via a piecewise linear approximation (triangulation) Such approximations are typically used in panel ....

K. E. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, Numerical Solution of Integral Equations (New York) (M. Goldberg, ed.), Plenum Press, 1990, pp. 1--34.

....fulfil this condition. Figure 1 shows one example. We conclude this subsection with a brief overview over the various existing numerical schemes for the approximate solution of Au = f . Probably the first method was the so called panel method, based on piecewise constant collocation (cf.[42, 26, 43, 3]) This method was proved to converge in supremum norm provided that P satisfies a certain condition introduced by Wendland in [42] Moreover, Kral and Wendland [28] showed that the panel method is stable for the case of certain rectangular domains P including those used in our tests. The proof ....

Atkinson, K.E.: A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in: Numerical solution of integral equations (ed. M.Golberg) Plenum Press, New York, 1990.

....D e . Let u 2 C 1 (D e ) C 2 (D e ) and assume that Deltau(P ) 0 at all P 2 D e . Then Z S u(Q) nQ dS(Q) jP Gamma Qj Gamma Z S u(Q) Delta nQ 1 jP Gamma Qj # dSQ = 8 : 4 Gamma Omega Gamma P ) u(P ) P 2 S 4 u(P ) P 2 D e (4. 2) see Atkinson[2]. In formula (4.2) Omega Gamma P ) denotes the interior solid angle at P 2 S, defined in Atkinson[4, p. 430] If S is smooth, then Omega Gamma P ) 2 . For a cube, the corners have interior solid angle of 1 2 , and the edges have interior solid angles of . To study the solvability of ....

K. Atkinson. A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions. Plenum Publishing, 1990.

.... method, e fN can be shown to satisfy kf Gamma e fN k1 = O( b ffi 3 N ) when S is a smooth surface; this is based on results from Nedelec[8] For piecewise smooth surfaces, it has been shown to be at least O( b ffi 2 N ) see Atkinson[3] But for the Nystr om method (see Atkinson[5]) kf Gamma b fN k1 Ck(K Gamma KN )fk1 : 12) Thus for the collocation method, we have the alternative error bound max 1iNv j f(v i ) Gamma e fN (v i ) j Ck(K Gamma KN )fk1 : 4 THE COLLOCATION METHOD 12 With this as motivation, we examine the error k(K Gamma KN )fk1 . Atkinson[3] has ....

Atkinson, K., A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, edited by M. Golberg, Plenum Press, New York, 1990.

....equations. In general, most work has been for the two dimensional problem; see Brebbia[13, 14, 15, 16, 17] All aspects of the three dimensional problem are less developed than for the two dimensional problem. Several methods have been used to study the three dimensional problem (see Atkinson[8]) and most numerical methods can be considered to be of collocation or Galerkin type. Atkinson[2, 6, 7] and Wendland[33, 36] used the collocation method, and Costabel and Stephan[22] Ervin and Stephan[23, 24] Giroire and Nedelec[25] and Wendland[35] studied Galerkin s method. Also, Nedelec[29] ....

Atkinson, K., A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, edited by M. Golberg, Plenum Press, New York, 1990.

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K. Atkinson (1990) A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Pub., New York, pp. 1-34.

....of this package is to allow for experimentation with numerical methods for solving boundary integral equations that are defined on piecewise smooth surfaces in R . General surveys of the numerical solution of boundary integral equation (BIE) reformulations of Laplace s equation are given in [5] and [9] and an introduction to the numerical solution of such BIE is given in [8, Chap. 9] Our package is restricted to triangulations which are uniform , as there are additional difficulties in dealing with meshes that are suitably graded. We have included a few routines to create graded ....

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990, pp. 1-34.

....of this package is to allow for experimentation with numerical methods for solving boundary integral equations that are defined on piecewise smooth surfaces in R 3 . General surveys of the numerical solution of boundary integral equation (BIE) reformulations of Laplace s equation are given in [5] and [9] and an introduction to the numerical solution of such BIE is given in [8, Chap. 9] Our package is restricted to triangulations which are uniform , as there are additional difficulties in dealing with meshes that are suitably graded. We have included a few routines to create graded ....

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990, pp. 1-34.

....edges and corners. 3. COLLOCATION ON SMOOTH SURFACES. To define numerical methods for solving (1) we follow closely the ideas used in defining boundary element methods for solving boundary integral equation reformulations of elliptic partial differential equations on regions in R 3 (e.g. see [2], 4, Chaps 5,9] 5] 12] In this section, we develop numerical methods for the case that S is a smooth surface (although it need not be connected) and we follow the notation of (13) 14) when considering S. In the following section, we extend the numerical theory to the cases where S is ....

K. Atkinson (1990) A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Pub., New York, pp. 1-34.

No context found.

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990, pp. 1-34.

.... purpose of this package is to allow for experimentation with numerical methods for solving boundary integral equations that are defined on piecewise smooth surfaces in R 3 : For a general survey of the numerical solution of boundary integral equation reformulations of Laplace s equation, see [5]. The package is restricted to triangulations which are uniform . But we are developing additional routines to allow for the use of graded meshes in solving boundary integral equations on surfaces for which the unknown density function has poor behaviour near edges and corners of the surface. 1.1 ....

K. Atkinson, A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations, ed. by M. Golberg, Plenum Press, New York, 1990, pp. 1-34.

No context found.

K. E. Atkinson, "A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions," in Numerical Solution of Integral Equations, M. Goldberg, Ed. New York: Plenum, 1990.

No context found.

K. Atkinson(1990) A survey of boundary integral equation methods for the numerical solution of Laplace's equation in three dimensions, in Numerical Solution of Integral Equations,ed. by M. Golberg, Plenum Press, pp. 1-34.

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