SIAM J. Scl. COMPUT. Vol. 15, No. 5, pp. 1083-1104, September 1994 () 1994 Society for Industrial and Applied Mathematics TWO-GRID ITERATION METHODS FOR LINEAR INTEGRAL EQUATIONS OF THE SECOND KIND ON PIECEWISE SMOOTH SURFACES IN 3, KENDALL E. ATKINSONt Abstract. The numerical solution of integral equations of the second kind on surfaces in ]3 often leads to large linear systems that must be solved by iteration. An especially important class of such equations is boundary integral equation (BIE) reformulations of elliptic partial differential equations; and, in this paper BIEs of the second kind are considered for Laplace’s equation. The numerical methods used are based on piecewise polynomial isoparametric interpolation over the surface and the surface is also approximated by such interpolation. Two-grid iteration methods are considered for (1) integral equations with a smooth kernel function, (2) BIEs over smooth surfaces, and (3) BIEs over piecewise smooth surfaces. In the last case, standard two-grid iteration does not perform well, and a modified two-grid iteration method is proposed and examined empirically. Key words, boundary integral equations, iteration methods AMS subject classifications, primary’65R20; secondary 65N99, 65F10, 35J05 1. Introduction. The numerical solution of boundary integral equations that are refor- mulations of partial differential equations in/I 3 often leads to the solution of very large systems of linear equations. At present, these linear systems are usually solved by iteration and, traditionally, a geometric series (also called a Neumann series) has been the basis of the iteration scheme. Here we consider two-grid methods for the solution of integral equations of the second kind. Two-grid methods usually converge faster than the geometric series and, in practical terms, they are usually comparable in speed to multigrid methods, while being simpler to program. As our test case for solving a BIE, we consider the exterior Neumann problem for Laplace’s equation: Au(A) O, A De, (1.1) u(P) f(p), P S, l)e u(A)--- O(IAI-1), IVU(A)I O(IA1-2) aslPI--+ The region De I3\/3, with D an open, bounded, and simply connected region, and S is the boundary of D. The unit normal at P S, directed into D, is denoted by v,. Using Green’s third identity, (1.2) 4rru(A) f(Q) IA QI dSQ- u(Q)vQ IA- QI dSQ, ADe. To find u on S, we solve the integral equation (1.3) fs 0 [ 1 ] dSQ+[2zr-f2(P)lu(P) 2zru(P) + u(Q)v IP- QI f(Q) IP- QI dSQ’ P e S. *Received by the editors August 6, 1991; accepted for publication (in revised form) July 25, 1993. This work was supported in part by National Science Foundation grant DMS-9003287. tDepartment of Mathematics, University of Iowa, Iowa City, Iowa 52242 (alzk+/-nson@matzh. uf_owa, edu). 1083 Downloaded 03/25/15 to 128.118.10.207. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php