A virtual element approximation for the modified transmission eigenvalues for natural materials

TL;DR

This paper develops a virtual element method (VEM) for the modified transmission eigenvalue problem arising in inverse scattering for natural materials. Because the artificial diffusivity is positive, the left-hand sesquilinear form is noncoercive, which the authors address using the $\mathds{T}$-coercivity framework to obtain well-posedness of the source problem. Using spectral approximation theory, they derive a priori error estimates for eigenfunctions and eigenvalues, showing that $|\lambda_j-\widehat{\lambda}_{j,h}|\lesssim h^{2\min(l,s)}$ and $\|{\bf U}_j-{\bf U}_{j,h}\|_{\mathbf{\mathds{V}}}\lesssim h^{\min(l,s)}$ under regularity $M(\lambda_j)\subset H^{1+s}(\Omega)\times H^{1+s}(\Omega)$. The method employs VE spaces with projections $\Pi_{l,E}^{\nabla}$ and $\Pi_{l,E}^{0}$ and stabilization, yielding convergence of the discrete operators $\mathcal{A}_h$ to $\mathcal{A}$ and effective eigenpair computations on polygonal meshes. Numerical experiments on multiple geometries corroborate the theory, showing robust and optimal convergence for natural materials with absorbing media and illustrating the practical utility of the approach in inverse scattering contexts.

Abstract

In this paper, we discuss a virtual element approximation for the modified transmission eigenvalue problem in inverse scattering for natural materials. In this case, due to the positive artificial diffusivity parameter in the considered problem, the sesquilinear form at the left end of the variational form is not coercive. We first demonstrate the well-posedness of the discrete source problem using the $\mathds{T}$-coercivity property, then provide the a priori error estimates for the approximate eigenspaces and eigenvalues, and finally reports several numerical examples. The numerical experiments show that the proposed method is effective

Figures (25)

• Figure 6: Error curves of the first four approximate eigenvalues on the uniform mesh of $\Omega_L$ using the VEM with degree $l=1$.
• Figure 7: The error curve of the first approximate eigenvalue obtained by using the VEM with degree $l=1$ on $\Omega_L$ with different parameter selections for $\sigma^E$ and $\tau^E$.
• Figure 8: The error curve of the first approximate eigenvalue obtained by using the VEM with degree $l=2$ on $\Omega_L$ with different parameter selections for $\sigma^E$ and $\tau^E$.
• Figure 9: Error curves of the first four approximate eigenvalues on the $(a)$-type mesh generation of $\Omega_C$ using the VEM with degree $l=1$.
• Figure : quadrilateral mesh $\mathcal{T}_h^1$.

Theorems & Definitions (15)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 3.1
• Lemma 3.4