The mathematical analysis of these methods is largely incomplete. The presentations in this minisymposium will both i) address recent advances in the mathematical analysis of these problems and ii) compare the computational efficiency and accuracy of different methods.
Mathematicians will find both new mathematics and an exposure to the insights and needs of modern engineers and scientists. Engineers will gain an improved appreciation of the mathematical issues related to these problems, and the questions that remain to be answered. The interaction of these groups should provide new ideas for old problems and new opportunities for interdisciplinary collaboration.
Students and recent graduates will be exposed to problems that are easy to describe and understand, and for which the mathematical issues are engaging and challenging.
Jacobo Bielak and Richard C. MacCamy
A set of tools is being developed for simulating earthquake ground motion in large basins by the finite element method on parallel computers. In this presentation we will describe the simple absorbing boundary conditions we have used for truncating the three-dimensional computational domain, and will illustrate their performance on the simulation of the response of San Fernando Valley to earthquake excitation.
Leszek F. Demkowicz
Two fundamental techniques for solving exterior Laplace and Helmholtz equations are discussed -- boundary element and coupled infinite/finite element diescretizations. The presentation focuses on convergence proofs for various versions of the methods. Theoretical results are illustrated with several numerical tests using $hp$ finite element discretizations. A particular emphasis will be placed on the convergence analysis for strong and weak coupling of finite/infinite element discretizations.
Dan Givoli and Igor Patlashenko
Two issues related to the use of Dirichlet-to-Neumann (DtN) maps for unbounded-domain problems are discussed. The first is the consistent localization of the DtN map, and the use of high-order boundary conditions and finite elements. The exact DtN nonlocal conditions and the localized conditions are compared, theoretically and experimentally, and error estimates are presented for both cases. The second issue is the use of DtN-type schemes for certain elliptic nonlinear problems, where the nonlinearity extends to infinity. Several such schemes are discussed, and numerical results are presented.
Marcus J. Grote and Joseph B. Keller
Scattering from a body of arbitrary shape immersed in an unbounded medium arises in many applications such as acoustics or electromagnetics. Finite difference and finite element methods can handle complicated geometries, inhomogeneous media, and nonlinearities. However, they require an artificial boundary, which surrounds the region of interest. This artificial boundary truncates the unbounded exterior domain in order to fit the infinite problem on a finite computer. As a consequence, nonreflecting boundary conditions must be imposed at the artificial boundary to ensure that it appears transparent to the propagating waves and does not generate any spurious reflections. We shall present an {\sl exact nonreflecting boundary condition} for the time dependent wave equation in three space dimensions. This boundary condition is local in time but nonlocal in space. It ensures that the solution inside the bounded domain coincides with the solution in the unbounded domain. Thus, we retain the flexibility of numerical methods based on the differential equation formulation of the problem, without introducing any errors at the artificial boundary.
Thomas Hagstrom
We develop a practical theory of boundary conditions at artificial boundaries for hyperbolic equations. This entails the derivation of concrete representations of exact conditions and a study of the convergence of sequences of approximate conditions. We also consider the direct implementation of the exact conditions. Numerical experiments are presented and compared with the theoretical predictions. A fundamental conclusion is that very accurate solutions can be cheaply obtained for problems with constant coefficients in the far-field.
This talk will cover recent results on boundary conditions for wave-like equations. This is a joint work with T. Hagstrom of The University of New Mexico. Talk will cover structure of far field solutions of wave equations. Using this structure, we present a theory for deriving easily implementable finite/infinite order boundary conditions. Extensions to problems governed by first order hyperbolic equations will be presented. Simulations for problems governed by compressible Euler equations will also be discussed.
Gregory A. Kriegsmann
We describe a hybrid method which will efficiently and accurately compute the electromagnetic waves scattered by a large slowly changing cavity. The cavity is constructed of a waveguide with a flanged opening at one end that couples it to free space, and a perfectly conducting wall or another flanged opening at the other end. The length of this structure $L >> \lambda$ and its walls are tapered on a scale length $H $ where $L >> H >> \lambda$. This problem crudely models the air intake of a jet engine. Its size and scaling make a direct numerical approach prohibitively expensive.
We use an adiabatic mode theory to construct an impedance condition which is applied at the plane $0<Z=Z_L<<L$. This condition analytically models the cavity physics in the region $Z>Z_L$ and is used in conjunction with a numerical scheme, such as finite differences, to solve the scattering problem on a much smaller domain $D$. This domain is made up of the portion of the waveguide given by $0<Z<Z_L$ and the hemispherical region $r<R$ centered at the flanged opening $Z=0$. A radiation boundary operator is applied at $r=R$ and the impedance condition at $Z=Z_L$. Examples will be presented.
Numerous local boundary conditions have been proposed for use as radiation boundary conditions for the Helmholtz equation in an unbounded domain. The analysis of a new class of boundary value problems for the Helmholtz equation will be presented. These results provide one means for comparing the effectiveness of different radiation boundary conditions. The dependence of these conclusions on the geometry of the computational domain is of particular interest. Numerical results will also be discussed.