Virtual Elements for the Transmission Eigenvalue Problem on Polytopal Meshes
David Mora, Iván Velásquez
Abstract
The transmission eigenvalue problem is a challenging model in the inverse scattering theory and has important applications in this topic. The aim of this paper is to analyze a $C^1$ virtual element method on polytopal meshes in $\mathbb{R}^d$ $(d=2,3)$ for solving a quadratic and non-self-adjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained by using the approximation theory for compact non-self-adjoint operators. Finally, a set of numerical tests illustrating the good performance of the virtual scheme are presented.
Topics & Concepts
Eigenvalues and eigenvectorsMathematicsPolygon meshEigenfunctionApplied mathematicsDivide-and-conquer eigenvalue algorithmQuadratic equationInverse iterationFinite element methodTransmission (telecommunications)Mathematical analysisGeometryComputer scienceQuantum mechanicsPhysicsTelecommunicationsThermodynamicsNumerical methods in inverse problemsElectromagnetic Scattering and AnalysisNumerical methods in engineering