A type of multigrid method based on the fixed-shift inverse iteration for the Steklov eigenvalue problem. (English) Zbl 1416.65425
Summary: For the Steklov eigenvalue problem, we establish a type of multigrid discretizations based on the fixed-shift inverse iteration and study in depth its a priori/a posteriori error estimates. In addition, we also propose an adaptive algorithm on the basis of the a posteriori error estimates. Finally, we present some numerical examples to validate the efficiency of our method.
MSC:
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
Software:
iFEMReferences:
| [1] | Bergman, S.; Schiffer, M., Kernel Functions and Elliptic Differential Equations in Mathematical Physics (1953), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0053.39003 |
| [2] | Conca, C.; Planchard, J.; Vanninathan, M., Fluids and Periodic Structures, 38 (1995), Chichester, UK: John Wiley & Sons, Chichester, UK · Zbl 0910.76002 |
| [3] | Bermúdez, A.; Rodríguez, R.; Santamarina, D., A finite element solution of an added mass formulation for coupled fluid-solid vibrations, Numerische Mathematik, 87, 2, 201-227 (2000) · Zbl 0998.76046 · doi:10.1007/s002110000175 |
| [4] | Armentano, M. G.; Padra, C., A posteriori error estimates for the Steklov eigenvalue problem, Applied Numerical Mathematics, 58, 5, 593-601 (2008) · Zbl 1140.65078 · doi:10.1016/j.apnum.2007.01.011 |
| [5] | Alonso, A.; Dello Russo, A., Spectral approximation of variationally-posed eigenvalue problems by nonconforming methods, Journal of Computational and Applied Mathematics, 223, 1, 177-197 (2009) · Zbl 1156.65094 · doi:10.1016/j.cam.2008.01.008 |
| [6] | Li, Q.; Lin, Q.; Xie, H., Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations, Applications of Mathematics, 58, 2, 129-151 (2013) · Zbl 1274.65296 · doi:10.1007/s10492-013-0007-5 |
| [7] | Li, Q.; Yang, Y., A two-grid discretization scheme for the Steklov eigenvalue problem, Journal of Applied Mathematics and Computing, 36, 1-2, 129-139 (2011) · Zbl 1220.65160 · doi:10.1007/s12190-010-0392-9 |
| [8] | Bi, H.; Yang, Y., A two-grid method of the non-conforming Crouzeix-Raviart element for the Steklov eigenvalue problem, Applied Mathematics and Computation, 217, 23, 9669-9678 (2011) · Zbl 1222.65121 · doi:10.1016/j.amc.2011.04.051 |
| [9] | Cao, L.; Zhang, L.; Allegretto, W.; Lin, Y., Multiscale asymptotic method for Steklov eigenvalue equations in composite media, SIAM Journal on Numerical Analysis, 51, 1, 273-296 (2013) · Zbl 1267.65172 · doi:10.1137/110850876 |
| [10] | Xie, H., A type of multilevel method for the Steklov eigenvalue problem, IMA Journal of Numerical Analysis, 34, 2, 592-608 (2014) · Zbl 1312.65178 · doi:10.1093/imanum/drt009 |
| [11] | Zhang, X.; Yang, Y.; Bi, H., Spectral method with the tensor-product nodal basis for the Steklov eigenvalue problem, Mathematical Problems in Engineering, 2013 (2013) · Zbl 1299.65281 · doi:10.1155/2013/650530 |
| [12] | Bi, H.; Li, H.; Yang, Y., An adaptive algorithm based on the shifted inverse iteration for the Steklov eigenvalue problem, Applied Numerical Mathematics, 105, 64-81 (2016) · Zbl 1382.65372 · doi:10.1016/j.apnum.2016.02.003 |
| [13] | Babuska, I.; Osborn, J.; Ciarlet, P. G.; Lions, J. L., Eigenvalue problems in finite element methods (part 1), Handbook of Numerical Analysis, 2, 640-787 (1991), North-Holland, The Netherlands: Elsevier Science, North-Holland, The Netherlands · Zbl 0712.65091 |
| [14] | Babuška, I.; Osborn, J. E., Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems, Mathematics of Computation, 52, 186, 275-297 (1989) · Zbl 0675.65108 · doi:10.2307/2008468 |
| [15] | Yang, Y.; Bi, H., Two-grid finite element discretization schemes based on shifted-inverse power method for elliptic eigenvalue problems, SIAM Journal on Numerical Analysis, 49, 4, 1602-1624 (2011) · Zbl 1236.65143 · doi:10.1137/100810241 |
| [16] | Yang, Y.; Bi, H.; Han, J.; Yu, Y., The shifted-inverse iteration based on the multigrid discretizations for eigenvalue problems, SIAM Journal on Scientific Computing, 37, 6, A2583-A2606 (2015) · Zbl 1327.65229 · doi:10.1137/140992011 |
| [17] | Ainsworth, M.; Oden, J. T., A Posteriori Error Estimation in Finite Element Analysis (2011), New York, NY, USA: Wiley-Interscience, New York, NY, USA |
| [18] | Babuška, I.; Rheinboldt, W. C., A-posteriori error estimates for the finite element method, International Journal for Numerical Methods in Engineering, 12, 10, 1597-1615 (1978) · Zbl 0396.65068 · doi:10.1002/nme.1620121010 |
| [19] | Morin, P.; Nochetto, R. H.; Siebert, K., Convergence of adaptive finite element methods, SIAM Review, 44, 4, 631-658 (2002) · Zbl 1016.65074 · doi:10.1137/s0036144502409093 |
| [20] | Chen, L., iFEM: an innovative finite element methods package in MATLAB (2009), Irvine, Calif, USA: University of California, Irvine, Calif, USA |
| [21] | Li, Y. J.; Yang, Y. D.; Bi, H., The spectral element method for the Steklov eigenvalue problem, Advanced Materials Research, 853, 631-635 (2014) · doi:10.4028/www.scientific.net/AMR.853.631 |
| [22] | Garau, E. M.; Morin, P., Convergence and quasi-optimality of adaptive FEM for Steklov eigenvalue problems, IMA Journal of Numerical Analysis, 31, 3, 914-946 (2011) · Zbl 1225.65107 · doi:10.1093/imanum/drp055 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
