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Multilevel local defect-correction method for the non-selfadjoint Steklov eigenvalue problems

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Abstract

In this paper, we design a multilevel local defect-correction method to solve the non-selfadjoint Steklov eigenvalue problems. Since the computation work needed for solving the non-selfadjoint Steklov eigenvalue problems increases exponentially as the scale of the problems increase, the main idea of our algorithm is to avoid solving large-scale equations especially large-scale Steklov eigenvalue problems directly. Firstly, we transform the non-selfadjoint Steklov eigenvalue problem into some symmetric boundary value problems defined in a multilevel finite element space sequence, and some small-scale non-selfadjoint Steklov eigenvalue problems defined in a low-dimensional auxiliary subspace. Next, the local defect-correction method is used to solve the symmetric boundary value problems, then the difficulty of solving these symmetric boundary value problems is further reduced by decomposing these large-scale problems into a series of small-scale subproblems. Overall, our algorithm can obtain the optimal error estimates with linear computational complexity, and the conclusions are proved by strict theoretical analysis which are different from the developed conclusions for equations with the Dirichlet boundary conditions.

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Funding

The Project is funded by Beijing Municipal Natural Science Foundation Grant No. 1232001, General projects of science and technology plan of Beijing Municipal Education Commission Grant No. KM202110005011, National Natural Science Foundation of China Grant nos. 11801021 and 12001402.

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Authors and Affiliations

  1. School of Mathematics, Statistics and Mechanics, Beijing University of Technology, Beijing, 100124, China

    Fei Xu & Bingyi Wang

  2. Center for Applied Mathematics and KL-AAGDM, Tianjin University, Tianjin, 300072, China

    Manting Xie

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  1. Fei Xu
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  2. Bingyi Wang
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Correspondence to Manting Xie.

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Xu, F., Wang, B. & Xie, M. Multilevel local defect-correction method for the non-selfadjoint Steklov eigenvalue problems. Bit Numer Math 64, 21 (2024). https://doi.org/10.1007/s10543-024-01022-z

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