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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

  • *Corresponding author: Jing An and Jianwei Zhou

    *Corresponding author: Jing An and Jianwei Zhou 

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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  • 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.

    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)

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

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

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

    Table 1.  The errors E(v, vMN) 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
     | Show Table
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    Table 2.  The first eigenvalue λmN for m = 0, 1, 2, 3 and different N.

    N λ0N λ1N λ2N λ3N
    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
     | Show Table
    DownLoad: CSV

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

    h λ0h λ1h λ2h λ3h
    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
     | Show Table
    DownLoad: CSV
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