High Frequency Wave Propagation

CSIC Building (#406), Seminar Room 4122.
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The Numerical Solution of Time-Harmonic Acoustic and Electromagnetic Scattering Problems by the Ultra Weak Variational Formulation

Peter Monk

Mathematical Sciences at University of Delaware

Abstract:  When electromagnetic or acoustic waves impinge on an object, they are scattered by reflection, refraction and other mechanisms depending on the shape and properties of the scatterer. We wish to compute the scattered field resulting from the interaction of an object (the scatterer) and a known incident field. Although such problems are usually linear and well-posed they are difficult to solve numerically because the oscillatory nature of the solution forces the use of large numbers of degrees of freedom in the numerical method, and the resulting linear system defies standard approaches such as multigrid. This is a particular problem at high frequencies when the scatterer spans many wavelengths. In an effort to improve the efficiency of a volume based approach as the frequency increases and to allow the solution of problems at widely different frequencies on a single grid, we have investigated the use of plane waves as a basis for approximating the scattered field. These are used in a discontinuous Galerkin scheme based on a tetrahedral finite element mesh. This method is termed the Ultra Weak Variational Formulation (UWVF) by its originators O. Cessenat and B. Despres. In joint work with Tomi Huttunen (Finland) and Eric Darrigrand (France) we have developed the UWVF by addressing certain conditioning problems that may arise if the plane waves are not chosen carefully. We have also shown how to use the Perfectly Matched Layer or Fast Multipole Method to improve the artificial boundary condition needed by the method. Interestingly the linear system from the UWVF is easier to solve than the one arising from the finite element method, and this allows a simple parallel implementation of the method. The method has been validated on a variety of problems, and extended to the acoustic-elastic fluid-structures problem. In the talk I shall show how the UWVF may be derived as a flux splitting discontinuous Galerkin method. I shall then show numerical and analytic results that demonstrate both the successes and drawbacks of the method applied to both ultrasonic simulations and also to electromagnetic scattering problems.

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