Hybrid Boundary Integral-Generalized (Partition of Unity) Finite-Element Solvers for the Scalar Helmholtz Equation

Finite-element-based techniques are one of the most popular methods used to model electromagnetic field behavior, but rely on the underlying tesselation to construct the ansatz space. Recently, Babuska and his colleagues developed the generalized finite-element method (GFEM), which overcomes this constraint and admits a larger class of basis functions. Application of this technique has been largely restricted to Poisson systems. In this paper, we explore the applicability of this technique to two-dimensional Helmholtz systems. We investigate methods necessary to impose various boundary conditions. We use this analysis to build the framework for hybridizing boundary integral techniques with GFEM, thus imposing an exact boundary condition to truncate the computational domain. We validate the results against analytical data for canonical geometries, and we demonstrate h and p convergence of this technique. Finally, to further validate the proposed approach for complex scatterers, we augment GFEM with perfectly matched layers, and compare it against the results obtained by using boundary integral GFEM