DMS Applied Mathematics Seminar

Speaker: Dr. Shelvean Kapita, University of Georgia

Title: Bivariate Spline Solutions to the Helmholtz Equation 

Abstract: Although there are many computational methods for solving the Helmholtz equation, e.g., \(hp\) finite element methods, the numerical solution of the Helmholtz equation still poses challenges, particularly for large wavenumbers. We shall explain how to use bivariate splines to numerically solve the Helmholtz equation in both bounded and unbounded domains. In addition, we shall establish existence, uniqueness and stability of the weak solution of the Helmholtz equation, under the assumption that \(k^2\), where \(k\) is the wavenumber, is not a Dirichlet eigenvalue of the associated Poisson equation. With this assumption, the standard assumption that the domain be strictly star-shaped  is no longer needed. Finally, we will explain how to use bivariate splines to solve the exterior domain Helmholtz equation using a PML technique. We demonstrate the effectiveness of bivariate splines for the bounded domain and exterior Helmholtz equation with a variety of numerical examples.