An H m -conforming spectral element method on multi-dimensional domain and its application to transmission eigenvalues

Abstract

We develop an H m -conforming (m ≥ 1) spectral element method on multi-dimensional domain associated with the partition into multi-dimensional rectangles. We construct a set of basis functions on the interval [−1, 1] that are made up of the generalized Jacobi polynomials (GJPs) and the nodal basis functions. So the basis functions on multi-dimensional rectangles consist of the tensorial product of the basis functions on the interval [−1, 1]. Then we construct the spectral element interpolation operator and prove the associated interpolation error estimates. Finally, we apply the H 2-conforming spectral element method to the Helmholtz transmission eigenvalues that is a hot problem in the field of engineering and mathematics.