RESUMEN

In this paper a new solution method for an inverse problem for the two-dimensional (2D) Helmholtz equation is developed. The underlying application area which motivated this work is the imaging of land mines using ground penetrating radar, formulated as an inverse problem for the Helmholtz equation. A second generation version of the elliptic systems method is developed to numerically solve this problem. Uniqueness and convergence theorems and a variety of numerical results are presented.

RESUMEN

This paper proves regularity estimates up to the boundary for solutions of elliptic systems of finite difference equations. The regularity estimates, obtained for boundary-fitted coordinate systems on domains with smooth boundary, involve discrete Sobolev norms and are proved using pseudodifference operators to treat systems with variable coefficients, The elliptic systems of difference equations and the boundary conditions that are considered are very general in form. It is proved that regularity of a regular elliptic system of difference equations is equivalent to the nonexistence of “eigensolutions.” The regularity estimates obtained are analogous to those in the theory of elliptic systems of partial differential equations, and to the results of Gustafsson, Kreiss, and Sundström [Math. Comp., 26 (1972), pp. 649–686] and others for hyperbolic difference equations.