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Fast Solution of Boundary Integral Equations by Using Multigrid Methods and Multipole Evaluation Techniques C

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Fast Solution of Boundary Integral Equations by Using Multigrid Methods and Multipole Evaluation Techniques C Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X Fast solution of boundary integral equations by using multigrid methods and multipole evaluation techniques C. Caspar Szechenyi htvdn College, Department of Mathematics P.O.Boz 70), #-2007 Oyo'r, #tm#an/ E-mail: [email protected] Abstract The standard Boundary Integral Equation Method generally results in dense and nonsymmetric algebraic equations. To speed up the computations, spe- cial techniques are needed. In this paper a multigrid method is presented applied to boundary integral equations. The main idea of the method is to convert a mixed boundary value problem to a sequence of pure Dirichlet and Neumann subproblems. To evaluate the appearing boundary integral operators, a special panel clustering method based on the fast multipole evaluation technique is applied. A completely different multigrid approach for solving the scattered data interpolation problem arising in the dual reci- procity method is also presented. 1 Introduction The usual discretisation techniques applied to boundary integral equations lead to an algebraic system with dense and non-self adjoint matrix even if the original problem was self adjoint in some Sobolev space. Therefore tra- ditional direct solvers (e.g. Gaussian elimination) are usually applied. The computational cost of these methods is proportional to the third power of the number of boundary nodes. In fact, the computational cost may exceed that of some advanced techniques applied to the original partial differential equation. For instance, if a standard multigrid method is applied to a 2D elliptic problem, the number of the necessary algebraic operations is pro- Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 604 Boundary Elements portional to the second power of the number of boundary gridpoints only, and can be reduced further by applying non-uniform grids. To speed up the solution of the boundary integral equations, several methods have been proposed. A natural attempt is the use of multigrid tools in boundary integral equation context. This technique is well known if the resulting boundary integral equation is of the second kind (Hackbusch [8]). However, if the original problem is supplied with mixed boundary con- ditions, this is not the case and the applicability of the multigrid method is not straightforward. A remedy is to convert the original (mixed) problem to a sequence of pure Dirichlet and Neumann subproblems, the solutions of which converging rapidly to the solution of the original problem as pro- posed by Caspar [3]. Here we present this method combined with the fast multipole technique of Rokhlin [13] to evaluate the appearing singular and hypersingular boundary integral operators. Using complex potentials, we derive the multipole expansions of these boundary integral operators. The technique uses a quadtree cell system controlled by the boundary and makes it possible to significantly reduce the computational cost of the appearing matrix-vector multiplications. The memory requirement is reduced as well, since it is not necessary to store the boundary element matrices. In many cases, the boundary integral equations contain also domain in- tegrals to be evaluated. The usual trick to evaluate these integrals is a transformation to boundary integrals. This can be carried out by using a (fast) Fourier transform (Tang [14]) or the multiple reciprocity method (Nowak and Brebbia [11]) or the dual reciprocity method of Nardini and Brebbia (see e.g. Partridge and Brebbia [12]). In this latter technique, us- ing also some internal points, the function to be integrated is approximated by simple radial basis functions, the integral of which can be transformed to the boundary in an easy way by applying Green's formulas. In fact, a scattered data interpolation problem is hidden in the background based on radial basis functions as pointed out by Golberg and Chen [6,7]. From many points of view the best choices for solving general scattered data in- terpolation problems have been proved Hardy's multiquadric method and the so-called thin plate spline method (see Franke [2] and Kansa [10]). Both methods are closely related with the biharmonic interpolation, where the interpolation function is assumed to satisfy the biharmonic equation except the interpolation points. However, these interpolation techniques generally produce dense, nonsymmetric and ill-conditioned algebraic systems. This causes numerical problems similar to the boundary integral equations. In this paper we present also a multigrid-based solution technique. The essen- tial idea is to solve the associated biharmonic problem, whose discretisation is completely independent of the original boundary integral equations and can be multigridded in a particularly efficient way using quadtree-generated nonuniform grids as proposed by Caspar and Simbierowicz [5]. Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 605 2 Multigrid solution of boundary integral equations As a model problem, consider the simplest 2D Laplace equation in a bounded and piecewise smooth domain ft supplied with mixed boundary conditions: A(/ = 0 (1) U\ri = uolri, -Q^lr* = fob (2) where F = <9Q, the boundary of H and PI, ^ form a disjoint decomposition of F. We assume that neither F% nor ^ is empty. UQ, VQ are given functions of the Sobolev space Jf^F) and #-*/2(F), respectively. Then (l)-(2) has a unique solution in the Sobolev space //*(H,A) := {U £ H*(tt) : A[/ £ 1/2(0)}. Equation (1) is equivalent to the corresponding boundary integral equation: TTU + Ku - Rv = 0 (3) and also to the normal derivative boundary integral equation: (4) Here u = t/|r, v — |^|r, and K, R denote the usual double-layer potential and the single-layer potential, respectively: '= /r while K* is the adjoint of K and Q is a hypersingular boundary integral operator defined as the normal derivative of the double layer potential. The coefficient of u in Equation (3) becomes the solid angle of the boundary where F is not smooth. The main idea of the multigrid approach mentioned in the introduction is to convert (l)-(2) to a sequence of pure Dirichlet and Neumann subprob- lem. To do this, let P be a (not necessarily orthogonal) projector of the closed subspace of the functions of H*/*(T) vanishing along PI. Then the operators PI := I — P, PI \— P* are also projectors in the spaces //*/^(F) and //~*^(P), respectively, and can be interpreted as certain extensions from FI to F and from F2 to F, respectively. Define the following iteration At/n+l/2 = 0, £/n+l/2|r =%" + ^l(%0 ~ %n) (5) or r 1 = 0, "** |r = where Un := t/n|r, Vn+i/2 := ^%T^~lr- Equation (5) is a pure Dirichlet, while Equation (6) is a pure Neumann problem. The iteration can be interpreted Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X 606 Boundary Elements as follows: in the first half-step (Equation (5)) the Dirichlet boundary con- dition along PI is exactly satisfied, while in the second half-step (Equation (6)) it is the Neumann condition along 1^2 that is exactly fulfilled. Intro- ducing the Dirichlet-to-Neumann operator J by Ju := -^, where At/ = 0 in 0 and f/|r •= u, the iteration (5)-(6) can be written in a simpler form: %n+i := J-X/ - P')JP^ + 6, 6 := J-'[?2i;o + (/ - ^2)^1^0], (7) Based on the concept of quasi-orthogonality, it can be shown that, under some conditions, the operator «/"*(/ — P*)JP is a contraction in the space //*/^(F), thus the iteration (7) is convergent. For details, see Caspar [4]. In the practice, PI, P% can be defined e.g. by the coarse-grid ap- proximation of the special mixed problems P^u := f/|r, where A/7 = 0, in ft, [/|r. - u|r,, f|r, = 0 and P,v := f |r, where At/ = 0, in 0, [/[PI — 0? §^|r2 = Hr2 The steps of the iteration (7) can be realized by solving pure Dirichlet and pure Neumann subproblems, or, equivalently, the corresponding boundary integral equations (3) and (4), which are of the second kind. This allows the use of standard multigrid tools. In this case, it is clear that the whole algorithm needs O(N*) operations only, where TV is the number of boundary nodes. The computational cost can be reduced further, if the appearing matrix-vector multiplications (after some usual discretisation) are performed in a "fast" way (requiring less than O(N^) operations). Such a general technique is the panel clustering method (Hack- busch and Nowak [9]). In the next section we show a special version of this method based on the multipole expansions. 3 Fast multipole method for evaluating boundary in- tegrals In the basic version of the fast multipole method (for details, see Rokhlin [13]; van Dommelen and Rundensteiner [1]) the function to be evaluated is expressed in terms of multipole series centered at the points z\, ..., ZN G C: *(*):= XX$,-(z) (8) .7 = 1 where the functions 3>j are of the form: ™ j) (9) * Z with given coefficients a,j and a^ (j = 1 , . , TV, r = 1,2,...). The evaluation is based on the shifting theorem which states that a multipole series centered at Zi can be transformed to another one centered at another point z^ in the following way: OO OO 1 Transactions on Modelling and Simulation vol 18, © 1997 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 607 where the new coefficients are defined by , , ao(zi - 22)" , ^ /r - 60 := GO, br := + Moreover, if the original multipole series is convergent outside a circle cen- tered at z\ with radius R\, the shifted one is also convergent outside the circle centered at z^ with radius R^ = R\ + \z\ — Zz\.
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