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

TL;DR

This paper introduces an $H^m$-conforming spectral element method for multi-dimensional domains, utilizing generalized Jacobi polynomials, and applies it to compute Helmholtz transmission eigenvalues, demonstrating its effectiveness.

Contribution

It develops a new $H^m$-conforming spectral element method with basis functions based on GJPs for multi-dimensional domains, and applies it to transmission eigenvalue problems.

Findings

Constructed basis functions using GJPs and nodal basis.

Proved interpolation error estimates for the spectral element method.

Applied the method to Helmholtz transmission eigenvalues with promising results.

Abstract

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.