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doi: 10.3934/dcdsb.2022021
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An efficient finite element method and error analysis for fourth order problems in a spherical domain

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China

*Corresponding author: Jing An

Received  November 2020 Revised  November 2021 Early access February 2022

Fund Project: 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)

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.

Citation: Na Peng, Jiayu Han, Jing An. An efficient finite element method and error analysis for fourth order problems in a spherical domain. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022021
References:
[1]

I. BabuskaJ. Osborn and J. Pitkaranta, Analysis of mixed methods using mesh dependent norms, Math. Comp., 35 (1980), 1039-1062.  doi: 10.1090/S0025-5718-1980-0583486-7.

[2]

S. Balasundaram and P. K. Bhattacharyya, A mixed finite element method for fourth order elliptic equations with variable coefficients, Comput. Math. Appl., 10 (1984), 245-256.  doi: 10.1016/0898-1221(84)90052-X.

[3]

B. Bialecki and A. Karageorghis, A Legendre spectral galerkin method for the biharmonic dirichlet problem, SIAM J. Sci. Comput., 22 (2000), 1549-1569.  doi: 10.1137/S1064827598342407.

[4]

S. ChenY. Yang and S. Mao, Anisotropic conforming rectangular elements for elliptic problems of any order, Appl. Numer. Math., 59 (2009), 1137-1148.  doi: 10.1016/j.apnum.2008.05.004.

[5]

E. H. Doha and A. H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58 (2008), 1224-1244.  doi: 10.1016/j.apnum.2007.07.001.

[6]

B. Guo and H. Jia, Spectral method on quadrilaterals, Math. Comp., 79 (2010), 2237-2264.  doi: 10.1090/S0025-5718-10-02329-X.

[7]

B. Guo and L. Wang, Error analysis of spectral method on a triangle, Adv. Comput. Math., 26 (2007), 473-496.  doi: 10.1007/s10444-005-7471-8.

[8]

R. Guo and Y. Xu, Semi-implicit spectral deferred correction method based on the invariant energy quadratization approach for phase field problems, Commun. Comput. Phys., 26 (2019), 87-113.  doi: 10.4208/cicp.OA-2018-0034.

[9]

J. HuZ. Shi and J. Xu, Convergence and optimality of the adaptive Morley element method, Numer. Math., 121 (2012), 731-752.  doi: 10.1007/s00211-012-0445-0.

[10]

J. HuangL. Guo and Z. Shi, Vibration analysis of Kirchhoff plates by the Morley element method, J. Comput. Appl. Math., 213 (2008), 14-34.  doi: 10.1016/j.cam.2006.12.026.

[11]

J. HuangZ. Shi and Y. Xu, Some studies on mathematical models for general elastic multi-structures, Sci. China Ser. A, 48 (2005), 986-1007.  doi: 10.1007/BF02879079.

[12]

J. HuangZ. Shi and Y. Xu, Finite element analysis for general elastic multi-structures, Sci. China Ser. A, 49 (2006), 109-129.  doi: 10.1007/s11425-005-0118-x.

[13]

T. KusanoM. Naito and C. A. Swanson, Radial entire solutions of even order semilinear elliptic equations, Canad. J. Math., 40 (1988), 1281-1300. 

[14]

P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 9-53.  doi: 10.1051/m2an/197509R100091.

[15]

M. LiX. Guan and S. Mao, New error estimates of the Morley element for the plate bending problems, J. Comput. Appl. Math., 263 (2014), 405-416.  doi: 10.1016/j.cam.2013.12.024.

[16]

F. Liu and J. Shen, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 38 (2015), 4564-4575.  doi: 10.1002/mma.2869.

[17]

J. LiuS. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.  doi: 10.1016/j.jmaa.2012.05.063.

[18]

S. MaoS. Nicaise and Z. Shi, Error estimates of Morley triangular element satisfying the maximal angle condition, Int. J. Numer. Anal. Model., 7 (2010), 639-655. 

[19]

L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aeronautical Quarterly, 19 (1968), 149-169.  doi: 10.1017/S0001925900004546.

[20]

M. T. O. Pimenta and S. H. M. Soares, Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289.  doi: 10.1016/j.jmaa.2012.01.039.

[21]

J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[22]

Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125.  doi: 10.1016/j.jde.2009.02.016.

[23]

D. Wells and J. Banks, Using p-refinement to increase boundary derivative convergence rates, Int. J. Numer. Anal. Model., 16 (2019), 891–924, Available from: arXiv: 1711.05922.

show all references

References:
[1]

I. BabuskaJ. Osborn and J. Pitkaranta, Analysis of mixed methods using mesh dependent norms, Math. Comp., 35 (1980), 1039-1062.  doi: 10.1090/S0025-5718-1980-0583486-7.

[2]

S. Balasundaram and P. K. Bhattacharyya, A mixed finite element method for fourth order elliptic equations with variable coefficients, Comput. Math. Appl., 10 (1984), 245-256.  doi: 10.1016/0898-1221(84)90052-X.

[3]

B. Bialecki and A. Karageorghis, A Legendre spectral galerkin method for the biharmonic dirichlet problem, SIAM J. Sci. Comput., 22 (2000), 1549-1569.  doi: 10.1137/S1064827598342407.

[4]

S. ChenY. Yang and S. Mao, Anisotropic conforming rectangular elements for elliptic problems of any order, Appl. Numer. Math., 59 (2009), 1137-1148.  doi: 10.1016/j.apnum.2008.05.004.

[5]

E. H. Doha and A. H. Bhrawy, Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials, Appl. Numer. Math., 58 (2008), 1224-1244.  doi: 10.1016/j.apnum.2007.07.001.

[6]

B. Guo and H. Jia, Spectral method on quadrilaterals, Math. Comp., 79 (2010), 2237-2264.  doi: 10.1090/S0025-5718-10-02329-X.

[7]

B. Guo and L. Wang, Error analysis of spectral method on a triangle, Adv. Comput. Math., 26 (2007), 473-496.  doi: 10.1007/s10444-005-7471-8.

[8]

R. Guo and Y. Xu, Semi-implicit spectral deferred correction method based on the invariant energy quadratization approach for phase field problems, Commun. Comput. Phys., 26 (2019), 87-113.  doi: 10.4208/cicp.OA-2018-0034.

[9]

J. HuZ. Shi and J. Xu, Convergence and optimality of the adaptive Morley element method, Numer. Math., 121 (2012), 731-752.  doi: 10.1007/s00211-012-0445-0.

[10]

J. HuangL. Guo and Z. Shi, Vibration analysis of Kirchhoff plates by the Morley element method, J. Comput. Appl. Math., 213 (2008), 14-34.  doi: 10.1016/j.cam.2006.12.026.

[11]

J. HuangZ. Shi and Y. Xu, Some studies on mathematical models for general elastic multi-structures, Sci. China Ser. A, 48 (2005), 986-1007.  doi: 10.1007/BF02879079.

[12]

J. HuangZ. Shi and Y. Xu, Finite element analysis for general elastic multi-structures, Sci. China Ser. A, 49 (2006), 109-129.  doi: 10.1007/s11425-005-0118-x.

[13]

T. KusanoM. Naito and C. A. Swanson, Radial entire solutions of even order semilinear elliptic equations, Canad. J. Math., 40 (1988), 1281-1300. 

[14]

P. Lascaux and P. Lesaint, Some nonconforming finite elements for the plate bending problem, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 9-53.  doi: 10.1051/m2an/197509R100091.

[15]

M. LiX. Guan and S. Mao, New error estimates of the Morley element for the plate bending problems, J. Comput. Appl. Math., 263 (2014), 405-416.  doi: 10.1016/j.cam.2013.12.024.

[16]

F. Liu and J. Shen, Stabilized semi-implicit spectral deferred correction methods for Allen-Cahn and Cahn-Hilliard equations, Math. Methods Appl. Sci., 38 (2015), 4564-4575.  doi: 10.1002/mma.2869.

[17]

J. LiuS. Chen and X. Wu, Existence and multiplicity of solutions for a class of fourth-order elliptic equations in $R^N$, J. Math. Anal. Appl., 395 (2012), 608-615.  doi: 10.1016/j.jmaa.2012.05.063.

[18]

S. MaoS. Nicaise and Z. Shi, Error estimates of Morley triangular element satisfying the maximal angle condition, Int. J. Numer. Anal. Model., 7 (2010), 639-655. 

[19]

L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aeronautical Quarterly, 19 (1968), 149-169.  doi: 10.1017/S0001925900004546.

[20]

M. T. O. Pimenta and S. H. M. Soares, Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl., 390 (2012), 274-289.  doi: 10.1016/j.jmaa.2012.01.039.

[21]

J. Shen, T. Tang and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Series in Computational Mathematics, 41. Springer, Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.

[22]

Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations, 246 (2009), 3109-3125.  doi: 10.1016/j.jde.2009.02.016.

[23]

D. Wells and J. Banks, Using p-refinement to increase boundary derivative convergence rates, Int. J. Numer. Anal. Model., 16 (2019), 891–924, Available from: arXiv: 1711.05922.

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
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
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
N$N = {4}$ $N = {8}$ $N = {16}$ $N = {32}$ $N = {64}$
$M = 0$ 1.9279 1.9910 2.0000 1.98801.9997
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
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
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