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An efficient finite element method and error analysis for fourth order problems in a spherical domain

  • *Corresponding author: Jing An

    *Corresponding author: Jing An

This work is supported by National Natural Science Foundation of China (Grant No. 12061023), Guizhou Provincial Education Department Foundation (Qianjiaohe No. KY[2018]041), and Guizhou Provincical Science and Technology Foundation (ZK[2021]012)

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  • An efficient finite element method for the fourth order problems in a spherical domain will be solved in this paper. Initially, we derive the necessary pole conditions with the intention of overcoming the difficulty of singularity introduced by spherical coordinate transformation. Then the original problem is transformed into a series of equivalent one-dimensional problems by utilizing spherical harmonic functions expansion. Secondly, we introduce some appropriate weighted Sobolev spaces and derive weak form and corresponding discrete form for each one-dimensional fourth order problem based on these pole conditions. In addition, we illustrate the error estimate of the approximate solutions by employing Lax-milgram lemma and approximation property of the cubic Hermite interpolation operator. Eventually, we present the algorithm in detail and show its efficiency through some numerical examples.

    Mathematics Subject Classification: 65N25, 65N30.


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  • Figure 1.  The figures of the numerical solution (left) and the exact solution (right) for $ N = 20 $, $ M = 12 $

    Figure 2.  The error figures between numerical solution and exact solution with $ N = 10 $, $ M = 6 $(left) and $ N = 25 $, $ M = 10 $(right), respectively

    Figure 3.  The figures of the numerical solution $ u_{0h}^{0} $ and the exact solution $ u_{0}^{0} $ for $ N = 10 $(left) and $ N = 25 $(right) with $ m = 0 $, respectively

    Figure 4.  The figures of the numerical solution (left) and the exact solution (right) for $ N = 20 $, $ M = 12 $

    Figure 5.  The error $ e(u_{Mh}(x, y, z), u(x, y, z)) $ figures between numerical solution and exact solution with $ N = 10 $, $ M = 6 $(left) and $ N = 25 $, $ M = 10 $(right), respectively

    Table 1.1.  The error $e(u_{Mh}(x, y, z), u(x, y, z))$ for different $M$ and $N$

    N $M = 2$ $M = 4$ $M = 6$ $M = 8$
    10 5.9016e-06 5.9016e-06 5.9041e-06 8.1147e-06
    15 1.1920e-06 1.1920e-06 1.1920e-06 1.1920e-06
    20 3.8196e-07 3.8196e-07 3.8196e-07 3.8196e-07
    25 1.5769e-07 1.5769e-07 1.5769e-07 1.5769e-07
     | Show Table
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    Table 1.2.  The convergence $r_h(\widetilde e)$ with $m = 0$

    N$N = {4}$ $N = {8}$ $N = {16}$ $N = {32}$ $N = {64}$
    $M = 0$ 1.9279 1.9910 2.0000 1.98801.9997
     | Show Table
    DownLoad: CSV

    Table 1.3.  The error $e(u_{Mh}(x, y, z), u(x, y, z))$ between numerical solutions and exact solution for different $M$ and $N$

    N$M = 2$ $M = 4$ $M = 6$ $M = 8$
    10 0.0163 4.3402e-04 8.9630e-04 0.0117
    15 0.0164 3.7736e-04 1.6157e-05 1.4130e-05
    20 0.0164 3.7154e-04 7.0821e-06 4.2228e-06
    25 0.0164 3.6885e-04 4.7106e-061.7312e-06
     | Show Table
    DownLoad: CSV
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