In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.
Accepté le :
Publié le :
DOI : 10.1051/m2an/2021039
Keywords: Non-selfadjoint steklov eigenvalue problem, full multigrid method, multilevel correction method, adaptive finite element method
@article{M2AN_2021__55_5_1779_0,
author = {Xie, Manting and Xu, Fei and Yue, Meiling},
title = {A type of full multigrid method for non-selfadjoint {Steklov} eigenvalue problems in inverse scattering},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1779--1802},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021039},
mrnumber = {4313373},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021039/}
}
TY - JOUR AU - Xie, Manting AU - Xu, Fei AU - Yue, Meiling TI - A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1779 EP - 1802 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021039/ DO - 10.1051/m2an/2021039 LA - en ID - M2AN_2021__55_5_1779_0 ER -
%0 Journal Article %A Xie, Manting %A Xu, Fei %A Yue, Meiling %T A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1779-1802 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021039/ %R 10.1051/m2an/2021039 %G en %F M2AN_2021__55_5_1779_0
Xie, Manting; Xu, Fei; Yue, Meiling. A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 1779-1802. doi: 10.1051/m2an/2021039
[1] and , A posteriori error estimates for the Steklov eigenvalue problem. Appl. Numer. Math. 58 (2008) 593–601. | MR | Zbl | DOI
[2] and , Eigenvalue Problems, edited by and , Vol. II. In: Handbook of numerical analysis, Finite element methods (Part 1), North-Holland, Amsterdam (1991) 641–787. | MR | Zbl | DOI
[3] and , An optimal order process for solving finite element equations. Math. Comput. 36 (1981) 35–51. | MR | Zbl | DOI
[4] and , A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem. Appl. Math. Comput. 217 (2011) 9669–9678. | MR | Zbl
[5] , and , Two-grid discretizations and a local finite element scheme for a non-selfadjoint Stekloff eigenvalue problem. Comput. Math. Appl. 79 (2020) 1895–1913. | MR | DOI
[6] and , Approximation of Steklov Eigenvalues of Non-selfadjoint Second Order Elliptic Operators, edited by , In: Mathematical foundations of the finite element method with applications to PDEs. Academic Press, New York (1972) 387–408. | MR | Zbl
[7] and , New convergence estimates for multigrid algorithms. Math. Comput. 49 (1987) 311–329. | MR | Zbl | DOI
[8] and , The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). | MR | Zbl | DOI
[9] , , and , Steklov eigenvalues in inverse scattering. SIAM J. Appl. Math. 76 (2016) 1737–1763. | MR | DOI
[10] and , A discontinuous Steklov problem with an application to water waves. J. Math. Anal. Appl. 69 (1979) 540–558. | MR | Zbl | DOI
[11] , , and , Multiscale asymptotic method for Steklov eigenvalue equations in composite media. SIAM J. Numer. Anal. 51 (2013) 273–296. | MR | Zbl | DOI
[12] and , An adaptive finite element eigenvalue solver of asymptotic quasi-Optimal computational complexity. SIAM J. Numer. Anal. 50 (2012) 1029–1057. | MR | Zbl | DOI
[13] , , and , Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. | MR | Zbl | DOI
[14] and , On the efficiency of adaptive finite element methods for elliptic problems with discontinous coefficients. SIAM J. Sci. Comput. 24 (2002) 443–462. | MR | Zbl | DOI
[15] , and , A full multigrid method for eigenvalue problems. J. Comput. Phys. 322 (2016) 747–759. | MR | DOI
[16] , The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
[17] , and , Adaptive discontinuous Galerkin methods for eigenvalue problems arising in incompressible fluid flows. SIAM J. Sci. Comput. 31 (2008) 4607–4632. | MR | Zbl | DOI
[18] , and , Convergence and optimal complexity of adaptive finite element eigenvalue computations. Numer. Math. 110 (2008) 313–355. | MR | Zbl | DOI
[19] and , A convergent adaptive method for elliptic eigenvalue problems. SIAM J. Numer. Anal. 47 (2009) 1067–1091. | MR | Zbl | DOI
[20] , Multi-Grid Methods and Applications. Vol.4 of: Computational Mathematics. Springer-Verlag, Berlin-Heidelberg (1985). | MR | Zbl
[21] , and , A cascadic multigrid method for eigenvalue problem. J. Comput. Math. 35 (2017) 56–72. | MR
[22] and , A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15 (2001) 107–138. | MR | Zbl | DOI
[23] , and , A Multilevel correction type of adaptive finite element method for eigenvalue problems. SIAM J. Sci. Comput. 40 (2018) A4208–A4235. | MR | DOI
[24] , and , A multilevel correction adaptive finite element method for Kohn-Sham equation. J. Comput. Phys. 355 (2018) 436–449. | MR | DOI
[25] , and , A multigrid method for Helmholtz transmission eigenvalue problems. J. Sci. Comput. 60 (2014) 276–294. | MR | Zbl | DOI
[26] , , and , A full multigrid method for nonlinear eigenvalue problems. Sci. China Math. 59 (2016) 2037–2048. | MR | DOI
[27] , , , , , and , The legacy of Vladimir Andreevich Steklov. Not. Am. Math. Soc. 61 (2014) 9–22. | MR | DOI
[28] and , Finite Element Methods: Accuracy and Inprovement. Science Press, Beijing (2006).
[29] and , A multi-level correction scheme for eigenvalue problems. Math. Comput. 84 (2015) 71–88. | MR | Zbl | DOI
[30] , and , An inverse medium problem using Stekloff eigenvalues and a Bayesian approach. Inverse Prob. 35 (2019). | MR
[31] , and , Spectral indicator method for a non-selfadjoint Steklov eigenvalue problem. J. Sci. Comput. 79 (2019) 1814–1831. | MR | DOI
[32] and , Discontinuous Galerkin methods of the non-selfadjoint Steklov eigenvalue problem in inverse scattering. Appl. Math. Comput. 381 (2020) 125–307. | MR
[33] , and , Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658. | MR | Zbl | DOI
[34] and , A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problem. Comput. Appl. Math. 62 (2011) 4100–4117. | MR | Zbl | DOI
[35] , A Review of a Posteriori Error Estimation Andadaptive Mesh-refinement Techniques. Wiley-Teubner, New York (1996). | Zbl
[36] , A multigrid method for eigenvalue problem. J. Comput. Phys. 274 (2014) 550–561. | MR | DOI
[37] , A type of multilevel method for the Steklov eigenvalue problem. IMA J. Numer. Anal. 34 (2014) 592–608. | MR | Zbl | DOI
[38] and , A multilevel correction method for interior transmission eigenvalue problem. J. Sci. Comput. 72 (2017) 586–604. | MR | DOI
[39] and , A multigrid method for the ground state solution of Bose-Einstein condensates. Commun. Comput. Phys. 19 (2016) 648–662. | MR | DOI
[40] and , Computable error estimates for ground state solution of Bose-Einstein condensates. J. Sci. Comput. 81 (2019) 1072–1087. | MR | DOI
[41] , , and , Computable error estimates for a nonsymmetric eigenvalue problem. East Asian J. Appl. Math. 7 (2017) 583–602. | MR | DOI
[42] and , A multilevel correction scheme for nonsymmetric eigenvalue problems by finite element methods. arXiv:1505.06288 (2015). | arXiv
[43] and , A multilevel finite element method for Fredholm integral eigenvalue problems. J. Comput. Phys. 303 (2015) 173–184. | MR | DOI
[44] , and , A parallel augmented subspace method for eigenvalue problems. SIAM J. Sci. Comput. 42 (2020) A2655–A2677. | MR | DOI
[45] and , Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73 (2003) 1139–1152. | MR | Zbl | DOI
[46] , , and , An asymptotically exact a posteriori error estimator for non-selfadjoint Steklov eigenvalue problem. Appl. Numer. Math. 156 (2020) 210–227. | MR | DOI
[47] , and , Error estimates and a two grid scheme for approximating transmission eigenvalues. arXiv:1506.06486v2 (2016). | arXiv
[48] , and , A cascadic multigrid method for nonsymmetric eigenvalue problem. Appl. Numer. Math. 146 (2019) 55–72. | MR | DOI
[49] and , A posteriori error estimates for a discontinuous Galerkin approximation of Steklov eigenvalue problems. Appl. Math. 62 (2017) 243–267. | MR | DOI
[50] , and , A multi-level mixed element method for the eigenvalue problem of biharmonic equation. J. Sci. Comput. 75 (2018) 1415–1444. | MR | Zbl | DOI
[51] , and , A multigrid correction scheme for a new Steklov eigenvalue problem in inverse scattering. Int. J. Comput. Math. (2019). DOI:. | DOI | MR | Zbl
[52] and , A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26 (2005) 1192–1213. | MR | Zbl | DOI
[53] and , The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique. Int. J. Numer. Methods Eng. 33 (1992) 1331–1364. | MR | Zbl | DOI
Cité par Sources :
