The solution of exterior interface problems using a variational method with Lagrange multipliers

Abstract

A numerical method is proposed and analyzed for approximately solving a class of exterior interface problems in three dimensions. The solution satisfies the Helmholtz equation near infinity as well as an outgoing or incoming radiation condition. The interface conditions involve jumps in both the function and the normal derivative of the function across a two-dimensional surface Γ. The former condition is an essential interface condition and is treated using Lagrange multipliers. The unboundedness of the exterior domain is treated by introducing an artificial sphere Γ R of sufficiently large radius R and an approximate local radiation boundary condition on this sphere. A variational formulation is obtained for this approximate problem that takes into account the approximate boundary condition on Γ R as well as the natural interface condition (i.e., the jump in the normal derivative on Γ). The resulting variational problem is discretized using the finite element method. It is proved that the discretization error is optimal in the sense of the approximation properties of the finite element subspaces. The discrete Lagrange multiplier formulation is generalized to take into account various perturbations of the method that often occur in its implementation. The error due to the perturbation is analyzed. It is also proved that the error due to the artificial boundary and approximate boundary condition is of order O ( R −2 ) for large R .