electromagnetics. The radiation boundary should be brought as close to the object or the scatter as possible. If one of these lower-order boundary conditions is used, then the needed accuracy is unavailable when the boundary is brought very close. So one must go to a higher-order boundary condition. A number of papers have looked at and developed higher-order boundary conditions. The challenge is how to implement those in an efficient way either into finite difference, finite element, or other discrete techniques for basic computational methods.

LESZEK DEMKOWICZ: I have two comments. First, I believe those high-order absorbing boundary conditions are nothing other than a truncated form of an infinite element. In a theory of infinite elements, it turns out that all one really needs is an extra infinite expansion in the fight direction. When second- or third-order truncated, nonreflecting boundary conditions are used, they actually correspond to a type of approximation in the real direction.

Second, when one looks at the classical separation-of-variables argument and the form of the exact solution, say, for the sphere problem under the Helmholtz equation, it becomes evident that the terms corresponding to a higher-order frequency on a sphere have simultaneously a larger exponent N in the denominator there. As one moves away from the body, the practical effect of those terms disappears, because having the denominator R raised to the power N for larger N makes the fraction converge to zero faster. For that reason and without any calculations, one can probably anticipate that there must be a tradeoff between the number of finite elements put in between the scatter and that artificial boundary (on which are positioned the infinite elements or nonreflecting boundary condition), and the numbers of terms in the real direction for the infinite element. The closer one gets to the scatter, the smaller will be the number of elements in between; but the number of terms in the infinite element will grow, and there is nothing that can be done about that. It is just the nature of the problem. Analogously, of course, as one gets further away from the scatter, then the number of the terms in the expansion one can use will probably be smaller, but that is compensated for by the number of elements that are positioned in between the two.

THOMPSON: I wish to respond to that. The infinite element and local radiation boundary conditions are both very similar in that they approximate the exact impedance on the radiation boundary. However, when one uses infinite elements and goes closer to the scatter, the fact that more terms or more layers in the infinite element must be used is offset by the cost [of adding extra layers for the infinite element] being much less than it is for having to discretize with regular finite elements in the domain that has been eliminated.

Also, the alternative to going to local boundary conditions and local infinite elements is, of course, to look at nonlocal boundary conditions that are theoretically exact. I am referring here to the Dirichlet-to-Neumann (DtN) boundary condition. It is nonlocal in this setting. So, if a direct solver were used, it would have disadvantages in storage costs. However, with some of the techniques for iterative solvers and some that Professor Pinsky presented—with matrix-free iterative solvers—one does not have to assemble a global matrix. There, some of the storage costs that arise from using nonlocal boundary conditions can be minimized or eliminated. So nonlocal boundary conditions require a very good approximation in order to go much higher up in the order of approximations that can be achieved.

HUGHES: I know that a number of acoustics techniques are already used in electromagnetics. Have infinite elements already been used also?

ZIOLKOWSKI: Yes, infinite elements have been used for a number of years.

HUGHES: It seems as though everything is used.

ZIOLKOWSKI: Yes, everything is used. What each technique is called for each community is going to be important with regard to interpretation. We in electromagnetics also call them infinite elements. My first project at Livermore was to do a global look-back scheme for the finite-difference time domain. It used Huygens' representation over a closed surface to provide the next point outside the surface in order to get an



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