An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering

Shixian Ren; Yu Zhang; Ziqiang Wang; Shixian Ren; Yu Zhang; Ziqiang Wang

doi:10.3934/math.2022423

AIMS Mathematics

2022, Volume 7, Issue 5: 7528-7551. doi: 10.3934/math.2022423

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An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering

1.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou 550025, China
2.
School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025, China
3.
School of Mathematics & Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou 550025, China

Received: 13 November 2021 Revised: 22 December 2021 Accepted: 30 December 2021 Published: 15 February 2022
MSC : 65N25, 65N35

An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. Firstly, we establish the weak form and the associated discrete scheme by introducing an appropriate Sobolev space and a corresponding approximation space. Then, according to the Fredholm Alternative, the corresponding operator forms of weak formulation and discrete formulation are derived. After that, the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators. We also construct an appropriate set of basis functions in the approximation space and derive the matrix form of the discrete scheme based on the tensor product. In addition, we extend the algorithm to the circular domain. Finally, we present plenty of numerical experiments and compare them with some existing numerical methods, which validate that our algorithm is effective and high accuracy.
- Steklov eigenvalue problem,
- spectral-Galerkin method,
- error estimations,
- tensor product,
- numerical experiments
Citation: Shixian Ren, Yu Zhang, Ziqiang Wang. An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering[J]. AIMS Mathematics, 2022, 7(5): 7528-7551. doi: 10.3934/math.2022423

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Abstract

An efficient spectral method is proposed for a new Steklov eigenvalue problem in inverse scattering. Firstly, we establish the weak form and the associated discrete scheme by introducing an appropriate Sobolev space and a corresponding approximation space. Then, according to the Fredholm Alternative, the corresponding operator forms of weak formulation and discrete formulation are derived. After that, the error estimates of approximated eigenvalues and eigenfunctions are proved by using the spectral approximation results of completely continuous operators and the approximation properties of orthogonal projection operators. We also construct an appropriate set of basis functions in the approximation space and derive the matrix form of the discrete scheme based on the tensor product. In addition, we extend the algorithm to the circular domain. Finally, we present plenty of numerical experiments and compare them with some existing numerical methods, which validate that our algorithm is effective and high accuracy.

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