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While scattering problems are posed on unbounded domains, volumetric discretizations typically require truncating the domain at a finite distance, closing the system with some sort of boundary condition. These conditions typically suffer from some deficiency, such as perturbing the boundary value problem to be solved or changing the character of the operator so that the discrete system is difficult to solve with iterative methods.
We introduce a new technique for the Helmholtz problem, based on using the Green formula representation of the solution at the artificial boundary. Finite element discretization of the resulting system gives optimal convergence estimates. The resulting algebraic system can be solved effectively with a matrix-free GMRES implementation, preconditioned with the local part of the operator. Extensions to the Morse-Ingard problem, a coupled system of pressure/temperature equations arising in modeling trace gas sensors, will also be given.