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Abstract
We investigate the application of multivariate splines (2D and 3D) to the Maxwell equations. Basic properties of spline functions and various traditional finite element formulations of the Maxwell equations for numerical analysis are reviewed. We find that a Helmholtz-type formulation is well suited for traditional node-based spline anal- ysis. Consequently, we study multivariate spline solutions to the Helmholtz equation with high wave number, a setting that poses numerical challenges which are well met by a new implementation of multivariate spline code.We extend this study to solve Maxwell boundary value problems in both poten- tial and Helmholtz-type formulations. We modify the traditional spline smoothness conditions to deal with domain inhomogeneities in a novel way. Our spline implemen- tation with arbitrary degree and modified smoothness conditions has the potential to address a variety of difficulties left unsolved by traditional nodal-based finite element methods.