The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem

Hao Li; Hai Bi; Yidu Yang

doi:10.3934/dcdsb.2020002

Article Contents

2020, Volume 25, Issue 5: 1775-1789. Doi: 10.3934/dcdsb.2020002

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The two-grid and multigrid discretizations of the $ C^0 $IPG method for biharmonic eigenvalue problem

School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

^* Corresponding author: Yidu Yang
^* Corresponding author: Yidu Yang

Received: April 30, 2018

Revised: July 31, 2019

Early access: December 2019

Published: May 2020

This work is supported by the National Natural Science Foundation of China (Grant No.11761022) and Science and Technology Foundation of Guizhou Province of China (Grant No. LH [2014] 7061)

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Abstract

In this paper, for the biharmonic eigenvalue problem with clamped boundary condition in $ \mathbb{R}^{2} $, we study the two-grid discretization based on shifted-inverse iteration of $ C^0 $IPG method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh $ \pi_h $ can be reduced to the solution of the eigenvalue problem on a coarser mesh $ \pi_H $ and the solution of a linear algebraic system on the fine mesh $ \pi_h $. We prove that the resulting solution still maintains an asymptotically optimal accuracy when $ h\geq O(H^3) $. In addition, we also discuss the multigrid discretization and the adaptive $ C^0 $IPG algorithm based on Rayleigh quotient iteration. Numerical experiments are provided to validate the theoretical analysis.

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Mathematics Subject Classification: Primary: 65N25, 65N30.

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Figure 1. The convergence rates for the unit square with a slit using quadratic(left) and cubic(right) $ C^0 $IPG methods

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Figure 2. The convergence rates for the L-shaped domain using quadratic(left) and cubic(right) $ C^0 $IPG methods

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Table 1. the first eigenvalue approximation for (2.1) on the unit square using quadratic $ C^0 $IPG method

$ H $	$ h $	$ \lambda_{H} $	$ t_1(s) $	$ \lambda^h $	$ t_2(s) $	$ \lambda_{h} $	$ t_3(s) $
$ \frac{\sqrt{2}}{8} $	$ \frac{\sqrt{2}}{32} $	1570.1117	.013	1307.7056	.033	1307.7017	.058
$ \frac{\sqrt{2}}{16} $	$ \frac{\sqrt{2}}{128} $	1350.9328	.044	1295.6742	1.01	1295.6742	1.61
$ \frac{\sqrt{2}}{32} $	$ \frac{\sqrt{2}}{512} $	1307.7017	.055	1294.9765	37.9	1294.9736	56.9
$ \frac{\sqrt{2}}{64} $	$ \frac{\sqrt{2}}{1024} $	1297.9745	.348	1294.8984	283	$ – $	$ – $

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Table 2. the first eigenvalue approximation for (2.1) on the unit square using cubic $ C^0 $IPG method

$ H $	$ h $	$ \lambda_{H} $	$ t_1(s) $	$ \lambda^h $	$ t_2(s) $	$ \lambda_{h} $	$ t_3(s) $
$ \frac{\sqrt{2}}{8} $	$ \frac{\sqrt{2}}{32} $	1300.7399	.017	1294.9569	.111	1294.9569	.226
$ \frac{\sqrt{2}}{16} $	$ \frac{\sqrt{2}}{128} $	1295.3239	.069	1294.9322	3.83	1294.9322	6.93
$ \frac{\sqrt{2}}{32} $	$ \frac{\sqrt{2}}{512} $	1294.9569	.222	1294.4529	208	1294.4621	1602

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Table 3. the first eigenvalue approximation for (2.1) on the unit square with a slit using quadratic $ C^0 $IPG method

$ H $	$ h $	$ \lambda_{H} $	$ t_1(s) $	$ \lambda^h $	$ t_2(s) $	$ \lambda_{h} $	$ t_3(s) $
$ \frac{\sqrt{2}}{8} $	$ \frac{\sqrt{2}}{32} $	10678.0553	.010	6541.4871	.037	6539.8146	.051
$ \frac{\sqrt{2}}{16} $	$ \frac{\sqrt{2}}{128} $	7228.7805	.011	6250.7646	.948	6250.6718	1.42
$ \frac{\sqrt{2}}{32} $	$ \frac{\sqrt{2}}{512} $	6539.8146	.070	6202.8719	29.7	6202.8660	44.0
$ \frac{\sqrt{2}}{64} $	$ \frac{\sqrt{2}}{1024} $	6328.9667	.298	6195.7389	217	$ -- $	$ -- $

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Table 4. the first eigenvalue approximation for (2.1) on the unit square with a slit using cubic $ C^0 $IPG method

$ H $	$ h $	$ \lambda_{H} $	$ t_1(s) $	$ \lambda^h $	$ t_2(s) $	$ \lambda_{h} $	$ t_3(s) $
$ \frac{\sqrt{2}}{8} $	$ \frac{\sqrt{2}}{32} $	6416.8901	.022	6226.2450	.096	6226.2445	.192
$ \frac{\sqrt{2}}{16} $	$ \frac{\sqrt{2}}{128} $	6271.1319	.032	6197.8931	3.24	6197.8930	5.91
$ \frac{\sqrt{2}}{32} $	$ \frac{\sqrt{2}}{512} $	6226.2445	.192	6190.6868	165	6190.6738	216

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Table 5. the first eigenvalue approximation for (2.1) on the L-shaped domain using quadratic $ C^0 $IPG method

$ H $	$ h $	$ \lambda_{H} $	$ t_1(s) $	$ \lambda^h $	$ t_2(s) $	$ \lambda_{h} $	$ t_3(s) $
$ \frac{\sqrt{2}}{8} $	$ \frac{\sqrt{2}}{32} $	11346.3219	.008	7019.4044	.017	7017.9070	.031
$ \frac{\sqrt{2}}{16} $	$ \frac{\sqrt{2}}{128} $	7713.4188	.008	6750.3026	.613	6750.2705	1.13
$ \frac{\sqrt{2}}{32} $	$ \frac{\sqrt{2}}{512} $	7017.9070	.032	6712.6852	21.8	6712.6814	32.3
$ \frac{\sqrt{2}}{64} $	$ \frac{\sqrt{2}}{1024} $	6818.6359	.204	6707.7302	155	6707.6827	191

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Table 6. the first eigenvalue approximation for (2.1) on the L-shaped domain using cubic $ C^0 $IPG method

$ H $	$ h $	$ \lambda_{H} $	$ t_1(s) $	$ \lambda^h $	$ t_2(s) $	$ \lambda_{h} $	$ t_3(s) $
$ \frac{\sqrt{2}}{8} $	$ \frac{\sqrt{2}}{32} $	6896.1972	.009	6729.3265	.066	6729.3263	.136
$ \frac{\sqrt{2}}{16} $	$ \frac{\sqrt{2}}{128} $	6764.7073	.022	6709.0991	2.35	6709.0990	4.16
$ \frac{\sqrt{2}}{32} $	$ \frac{\sqrt{2}}{512} $	6729.3263	.136	6704.3362	117	6704.3206	157

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