A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

Jiantao Jiang; Jing An; Jianwei Zhou

doi:10.3934/dcdsb.2022066

Article Contents

2023, Volume 28, Issue 1: 50-69. Doi: 10.3934/dcdsb.2022066

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A novel numerical method based on a high order polynomial approximation of the fourth order Steklov equation and its eigenvalue problems

1.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China
2.
School of Mathematics and Statistics, Linyi University, Linyi, 276005, China

^*Corresponding author: Jing An and Jianwei Zhou
^*Corresponding author: Jing An and Jianwei Zhou

Received: November 30, 2021

Revised: January 31, 2022

Early access: March 2022

Published: January 2023

This work is supported by the National Natural Science Foundation of China (Grant No.12061023), and Guizhou Provincial Education Department Foundation (Qianjiaohe No.KY[2018]041)

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Abstract

Based on high order polynomial approximation and dimension reduction technique, we propose a novel numerical method for the fourth order Steklov problems in the circular domain. We first decompose the primal problem into a set of 1D problems via polar coordinate transformation and Fourier basis functions expansion. Then, by introducing a non-uniformly weighed Sobolev space, the variational form and corresponding discrete scheme are derived. Employing the Lax-Milgram lemma and approximation properties of the projection operators, we further prove existence and uniqueness of weak solutions and approximation solutions for each one-dimensional problems, and the error estimation between them, respectively. We also carry out ample numerical experiments which illustrate that the numerical algorithm is efficient and highly accurate.

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Mathematics Subject Classification: 65N15, 65N25, 65N35.

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Figure 1. The comparison between approximation solutions $ v_{MN} $ for $ N = 30,M = 15 $(left), $ N = 50, M = 25 $(center), and reference solution(right)

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Figure 2. The error between reference solution and approximation solutions $ v_{MN} $ for $ N = 30,M = 15 $(left), $ N = 50 $ and $ M = 25 $(right)

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Figure 3. The error tendency for the fixed $ M $(left) and the fixed $ N $(right)

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Figure 4. The error between exact solution and approximation solution for $ N = 30 $(left), $ N = 50 $(right)

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Table 1. The errors E(v, v_MN) with different N and M.

N	M=5	M=10	M=15	M=20	M=25
10	9.6887e-07	1.0025e-04	1.0637e-04	1.0676e-04	1.0685e-04
20	5.4884e-08	5.6948e-12	4.7867e-08	9.1367e-06	1.0056e-05
30	5.4884e-08	4.7638e-13	1.2520e-17	7.9545e-13	9.1848e-09
40	5.4884e-08	4.7638e-13	1.0309e-17	1.0308e-17	1.0367e-17
50	5.4884e-08	4.7638e-13	9.0532e-18	9.0513e-18	9.0513e-18

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Table 2. The first eigenvalue λ_mN for m = 0, 1, 2, 3 and different N.

N	λ_0N	λ_1N	λ_2N	λ_3N
4	1.999999999999998	4.000000000000001	5.999999999999999	8.059210526315789
6	1.999999999999999	3.999999999999998	5.999999999999998	7.999999999999998
8	2.000000000000000	4.000000000000001	6.000000000000001	8.000000000000004
10	2.000000000000001	4.000000000000001	6.000000000000003	7.999999999999993
12	2.000000000000001	4.000000000000001	6.000000000000002	8.000000000000007

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Table 3. The first eigenvalue λ_mN for m = 0, 1, 2, 3 and different h.

h	λ_0h	λ_1h	λ_2h	λ_3h
1/4	2.000000032363347	4.000000018720217	6.000436469604340	8.005030682333810
1/6	2.000000028811607	4.000000048950857	6.000081442394085	8.000975861078535
1/8	2.000000024461073	4.000000073005518	6.000025208583129	8.000307091454056
1/10	2.000000022604670	4.000000056975401	6.000010214870977	8.000125489065235
1/12	2.000000020581500	4.000000061763554	6.000004896294027	8.000060442986863

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