Xia Ji

Xia Ji, Peijun Li, Jiguang Sun

The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear, non-self-adjoint, and of fourth order. We construct a nonlinear function whose values are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. Using an $H^2$-conforming finite element for the self-adjoint fourth order eigenvalue problems, we employ a secant method to compute the roots of the nonlinear function. The convergence of the proposed method is proved. In addition, a mixed finite element method is developed for the purpose of verification. Numerical examples are presented to verify the theory and demonstrate the effectiveness of the two methods.

Shuo Zhang, Yingxia Xi, Xia Ji

In this paper, we discuss approximating the eigenvalue problem of biharmonic equation. We first present an equivalent mixed formulation which admits amiable nested discretization. Then, we construct multi-level finite element schemes by implementing the algorithm as in [33] to the nested discretizations on series of nested grids. The multi-level mixed scheme for biharmonic eigenvalue problem possesses optimal convergence rate and optimal computational cost. Both theoretical analysis and numerical verifications are presented.

Yingxia Xi, Xia Ji

The goal of this paper is to develop numerical methods computing a few smallest elastic interior transmission eigenvalues, which are of practical importance in inverse elastic scattering theory. The problem is challenging since it is nonlinear, nonselfadjoint, and of fourth order. In this paper, we construct a lowest order mixed finite element method which is close to the Ciarlet-Raviart mixed finite element method. This scheme is based on Lagrange finite elements and is one of the less expensive methods in terms of the amount of degrees of freedom. Due to the nonselfadjointness, the discretization of elastic transmission eigenvalue problem leads to a non-classical mixed method which does not fit into the framework of classical theoretical analysis. In stead, we obtain the convergence analysis based on the spectral approximation theory of compact operators. Numerical examples are presented to verify the theory. Both real and complex eigenvalues can be obtained.

Dandan Chen Kaptur, Yue Huang, Xuejun Ryan Ji et al.

This research delved into GPT-4 and Kimi, two Large Language Models (LLMs), for systematic reviews. We evaluated their performance by comparing LLM-generated codes with human-generated codes from a peer-reviewed systematic review on assessment. Our findings suggested that the performance of LLMs fluctuates by data volume and question complexity for systematic reviews.

Y. Xi, X. Ji, S. Zhang

In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangle grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal com- putational cost can be obtained with the scheme.