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

In this paper we develop an $H^m$-conforming ($m\ge1$) 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 is 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 topic in the field engineering and mathematics.