arXiv subjects

Shixi Wang

Publications and source records attributed to Shixi Wang.

5 recordsLinked to original sources

arXiv2026

A high order stabilization-free virtual element method for general second-order elliptic eigenvalue problem

In this paper, we discuss a novel higher-order stabilization-free virtual element method for general second-order elliptic eigenvalue problems. Optimal a priori error estimates are derived for both the approximate eigenspace and eigenvalues. Numerical experiments are conducted on regular convex polygonal meshes, convex-concave polygonal meshes, and concave polygonal meshes. The numerical results validate the effectiveness of the proposed method.

math.NA↗

arXiv2026

Adaptive FEM with optimal convergence rate for non-self-adjoint eigenvalue problems

In this paper, we first discuss the optimal convergence of the adaptive finite element methods for non-self-adjoint eigenvalue problems. We present new theoretical error estimators and computable error estimators for multiple and clustered eigenvalues with the help of the error estimators of finite element solutions for the corresponding source problems, and prove the equivalence between these two estimators. We propose an adaptive algorithm for the eigenvalue cluster and demonstrate that it achieves the optimal convergence rate.We also provide numerical experiments to support our theoretical findings.

math.NA↗

arXiv2025

The discontinuous Galerkin method for the Oseen eigenvalue problem

In this paper, we focus on investigating symmetric and nonsymmetric discontinuous Galerkin (DG) methods for solving the Oseen eigenvalue problem based on the velocity-pressure formulation in $\mathbb{R}^{d}(d=2,3)$. We derive the a priori and a posteriori error estimates for the approximate eigenpairs for each method. We establish an adjoint-consistent symmetric DG method and derive optimal a priori error estimates, and prove the reliability and effectiveness of the error estimators for approximate eigenfunctions, as well as the reliability of the estimator for approximate eigenvalues. Numerical experiments confirm our theoretical analysis and demonstrate that the symmetric DG method achieves the optimal order of convergence, and that the nonsymmetric DG methods produce fewer spurious eigenvalues than the symmetric DG method for a fixed small penalty parameter $\gamma$.

math.NA↗

arXiv2025

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

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 report several numerical examples. The numerical experiments show that the proposed method is effective.

math.NA↗

arXiv2022

The a posteriori error estimates and an adaptive algorithm of the FEM for transmission eigenvalues for anisotropic media

The transmission eigenvalue problem arising from the inverse scattering theory is of great importance in the theory of qualitative methods and in the practical applications. In this paper, we study the transmission eigenvalue problem for anisotropic inhomogeneous media in $Ω\subset \mathbb{R}^d$,(d=2,3). Using the T-coercivity and the spectral approximation theory, we derive an a posteriori estimator of residual type and prove its effectiveness and reliability for eigenfunctions. In addition, we also prove the reliability of the estimator for transmission eigenvalues. The numerical experiments indicate our method is efficient and can reach the optimal order $DoF^{-2m/d}$ by using piecewise polynomials of degree $m$ for real eigenvalues.

math.NA↗