A finite element method for exterior interface problems

Abstract

A procedure is given for the approximate solution of a class of two-dimensional diffraction problems. Here the usual inner boundary conditions are replaced by an inner region together with interface conditions. The interface problem is treated by a variational procedure into which the infinite region behavior is incorporated by the use of a non-local boundary condition over an auxiliary curve. The variational problem is formulated and existence of a solution established. Then a corresponding approximate variational problem is given and optimal convergence results established. Numerical results are presented which confirm the convergence rates.

Keywords

• approximation property
• approximate variational problem
• convergence
• convergence rate
• elliptic
• finite elements
• Galerkin
• Helmholtz equation
• integral equation
• optimality
• potential theory
• regularity.