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April  2022, 15(4): 851-876. doi: 10.3934/dcdss.2022016

Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

Yixuan Wu and  Yanzhi Zhang ,

Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA

* Corresponding author: Yanzhi Zhang

Received  March 2021 Revised  December 2021 Published  April 2022 Early access  February 2022

Fund Project: This work was partially supported by the US National Science Foundation under Grant Number DMS-1913293 and DMS-1953177

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $ (- \Delta)^\frac{{ \alpha}}{{2}} $ for $ \alpha \in (0, 2) $. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of $ {\mathcal O}(h^2) $, while $ {\mathcal O}(h^4) $ for quadratic basis functions with $ h $ a small mesh size. This accuracy can be achieved for any $ \alpha \in (0, 2) $ and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies $ u \in C^{m, l}(\bar{ \Omega}) $ for $ m \in {\mathbb N} $ and $ 0 < l < 1 $, our method has an accuracy of $ {\mathcal O}(h^{\min\{m+l, \, 2\}}) $ for constant and linear basis functions, while $ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $ for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

Keywords: Fractional Laplacian, operator factorization, Lagrange basis functions, fractional Poisson problems, tempered fractional Laplacian.
Mathematics Subject Classification: Primary: 65R20, 65N35, 45E05; Secondary: 47G10, 47G30.
Citation: Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016
References:
[1]

G. Acosta, F. M. Bersetche and J. P. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl., 74 (2017), 784-816.  doi: 10.1016/j.camwa.2017.05.026.

[2]

G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), 472-495.  doi: 10.1137/15M1033952.

[3]

G. Acosta, J. P. Borthagaray and N. Heuer, Finite element approximations of the nonhomogeneous fractional Dirichlet problem, IMA J. Numer. Anal., 39 (2019), 1471-1501.  doi: 10.1093/imanum/dry023.

[4]

M. Ainsworth and C. Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary Computational Mathematics–-a Celebration of the 80th Birthday of Ian Sloan, Vol. 1, 2, 17–57, Springer, Cham, 2018.

[5]

U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.

[6]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80.

[7]

A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal., 37 (2017), 1245-1273.  doi: 10.1093/imanum/drw042.

[8]

J. Burkardt, Y. Wu and Y. Zhang, A unified meshfree pseudospectral method for solving both classical and fractional PDEs, SIAM J. Sci. Comput., 43 (2021), A1389–A1411. doi: 10.1137/20M1335959.

[9]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.

[10]

S. Duo, H. W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., 355 (2018), 233-252.  doi: 10.1016/j.jcp.2017.11.011.

[11]

S. Duo, H. Wang and Y. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110.

[12]

S. Duo and Y. Zhang, Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well, Commun. Comput. Phys., 18 (2015), 321-350.  doi: 10.4208/cicp.300414.120215a.

[13]

S. Duo and Y. Zhang, Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications, Comput. Methods. Appl. Mech. Eng., 355 (2019), 639-662.  doi: 10.1016/j.cma.2019.06.016.

[14]

S. Duo and Y. Zhang, Numerical approximations for the tempered fractional Laplacian: Error analysis and applications, J. Sci. Comput., 81 (2019), 569-593.  doi: 10.1007/s10915-019-01029-7.

[15]

Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference–quadrature approach, SIAM J. Numer. Anal., 52 (2014), 3056-3084.  doi: 10.1137/140954040.

[16]

F. Izsák and B. J. Szekeres, Models of space-fractional diffusion: A critical review, Appl. Math. Lett., 71 (2017), 38-43.  doi: 10.1016/j.aml.2017.03.006.

[17]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Phys. D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.

[18]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[19]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.

[20]

A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth and G. E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020), 109009.  doi: 10.1016/j.jcp.2019.109009.

[21]

X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508.  doi: 10.1016/j.crma.2012.05.011.

[22]

J. A. Rosenfeld, S. A. Rosenfeld and W. E. Dixon, A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions, J. Comput. Phys., 390 (2019), 306-322.  doi: 10.1016/j.jcp.2019.02.015.

[23]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.

[24]

C. Sheng, J. Shen, T. Tang, L.-L. Wang and H. Yuan, Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains, SIAM J. Numer. Anal., 58 (2020), 2435-2464.  doi: 10.1137/19M128377X.

[25]

T. Tang, L.-L. Wang, H. Yuan and T. Zhou, Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains, SIAM J. Sci. Comput., 42 (2020), A585–A611. doi: 10.1137/19M1244299.

[26]

Y. Wu and Y. Zhang, A universal solution scheme for fractional and classical PDEs, arXiv: 2102.00113, 2021.

[27]

Y. Wu and Y. Zhang, Variable-order Laplacian and its computations with meshfree methods, preprint, 2021.

show all references

References:
[1]

G. Acosta, F. M. Bersetche and J. P. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian, Comput. Math. Appl., 74 (2017), 784-816.  doi: 10.1016/j.camwa.2017.05.026.

[2]

G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM J. Numer. Anal., 55 (2017), 472-495.  doi: 10.1137/15M1033952.

[3]

G. Acosta, J. P. Borthagaray and N. Heuer, Finite element approximations of the nonhomogeneous fractional Dirichlet problem, IMA J. Numer. Anal., 39 (2019), 1471-1501.  doi: 10.1093/imanum/dry023.

[4]

M. Ainsworth and C. Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, Contemporary Computational Mathematics–-a Celebration of the 80th Birthday of Ian Sloan, Vol. 1, 2, 17–57, Springer, Cham, 2018.

[5]

U. Biccari, M. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.

[6]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80.  doi: 10.4064/sm-123-1-43-80.

[7]

A. Bonito and J. E. Pasciak, Numerical approximation of fractional powers of regularly accretive operators, IMA J. Numer. Anal., 37 (2017), 1245-1273.  doi: 10.1093/imanum/drw042.

[8]

J. Burkardt, Y. Wu and Y. Zhang, A unified meshfree pseudospectral method for solving both classical and fractional PDEs, SIAM J. Sci. Comput., 43 (2021), A1389–A1411. doi: 10.1137/20M1335959.

[9]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints, SIAM Rev., 54 (2012), 667-696.  doi: 10.1137/110833294.

[10]

S. Duo, H. W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., 355 (2018), 233-252.  doi: 10.1016/j.jcp.2017.11.011.

[11]

S. Duo, H. Wang and Y. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110.

[12]

S. Duo and Y. Zhang, Computing the ground and first excited states of the fractional Schrödinger equation in an infinite potential well, Commun. Comput. Phys., 18 (2015), 321-350.  doi: 10.4208/cicp.300414.120215a.

[13]

S. Duo and Y. Zhang, Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications, Comput. Methods. Appl. Mech. Eng., 355 (2019), 639-662.  doi: 10.1016/j.cma.2019.06.016.

[14]

S. Duo and Y. Zhang, Numerical approximations for the tempered fractional Laplacian: Error analysis and applications, J. Sci. Comput., 81 (2019), 569-593.  doi: 10.1007/s10915-019-01029-7.

[15]

Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: A finite difference–quadrature approach, SIAM J. Numer. Anal., 52 (2014), 3056-3084.  doi: 10.1137/140954040.

[16]

F. Izsák and B. J. Szekeres, Models of space-fractional diffusion: A critical review, Appl. Math. Lett., 71 (2017), 38-43.  doi: 10.1016/j.aml.2017.03.006.

[17]

K. Kirkpatrick and Y. Zhang, Fractional Schrödinger dynamics and decoherence, Phys. D, 332 (2016), 41-54.  doi: 10.1016/j.physd.2016.05.015.

[18]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.

[19]

N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.

[20]

A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth and G. E. Karniadakis, What is the fractional Laplacian? A comparative review with new results, J. Comput. Phys., 404 (2020), 109009.  doi: 10.1016/j.jcp.2019.109009.

[21]

X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508.  doi: 10.1016/j.crma.2012.05.011.

[22]

J. A. Rosenfeld, S. A. Rosenfeld and W. E. Dixon, A mesh-free pseudospectral approach to estimating the fractional Laplacian via radial basis functions, J. Comput. Phys., 390 (2019), 306-322.  doi: 10.1016/j.jcp.2019.02.015.

[23]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.

[24]

C. Sheng, J. Shen, T. Tang, L.-L. Wang and H. Yuan, Fast Fourier-like mapped Chebyshev spectral-Galerkin methods for PDEs with integral fractional Laplacian in unbounded domains, SIAM J. Numer. Anal., 58 (2020), 2435-2464.  doi: 10.1137/19M128377X.

[25]

T. Tang, L.-L. Wang, H. Yuan and T. Zhou, Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains, SIAM J. Sci. Comput., 42 (2020), A585–A611. doi: 10.1137/19M1244299.

[26]

Y. Wu and Y. Zhang, A universal solution scheme for fractional and classical PDEs, arXiv: 2102.00113, 2021.

[27]

Y. Wu and Y. Zhang, Variable-order Laplacian and its computations with meshfree methods, preprint, 2021.

Figure 1.  Comparison of our method with linear basis (i.e., $ \varphi^1 $) or quadratic basis (i.e., $ \varphi^2 $) and the method in [15] with linear basis (i.e., Huang2014) in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u(x) $, where $ u(x) = (1-x^2)^6_+ $ and the error function is defined in (28). (a) $ \alpha = 0.6 $; (b) $ \alpha = 1.5 $
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Figure 2.  Numerical error $ |e_ \Delta(x)| $ at point $ x = 0 $ (left) and $ x = 0.5 $ (right) in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u $ with $ u(x) = (1-x^2)_+^{1+\lfloor \alpha\rfloor} $, where $ \alpha = 0.5 $ (blue), $ 1 $ (red), and $ 1.7 $ (green)
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Figure 3.  Numerical errors $ \|e_ \Delta\|_\infty $ in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u $, where $ \alpha = 0.5 $ (blue), $ 1 $ (red), and $ 1.7 $ (green). (a) $ u = (1-x^2)_+^{3.1+ \alpha} $; (b) $ u = (1-x^2)_+^{4.1+ \alpha} $
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Figure 4.  Comparison of our method with linear basis (i.e., $ \varphi^1 $) or quadratic basis (i.e., $ \varphi^2 $) and the finite difference method in [10] (i.e., FDM) in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u(x) $, where $ u(x) = (1-x^2)^{4.1+ \alpha}_+ $, and $ \alpha = 0.5 $ (a) or $ 1.7 $ (b)
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Figure 5.  Effects of splitting parameter $ \gamma $ in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u $ on $ (-1, 1) $ with $ u(x) = 1/(1+x^2) $, where $ \alpha = 0.5 $ (blue), $ 1 $ (red), and $ 1.7 $ (green)
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Figure 6.  Numerical errors $ |e_u(x)| $ at point $ x = 0 $ in solving the 1D fractional Poisson problem (1)–(2) with $ f(x) = 1 $ and $ g(x) = 0 $, where $ \alpha = 0.5 $ (blue), $ 1 $ (red), or $ 1.7 $ (green)
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Figure 7.  Numerical errors $ \|e_u\|_\infty $ in solving the 1D Poisson problem (1)–(2) with $ g(x) = 0 $ and $ f(x) $ in (29), where the exact solution is $ u(x) = (1-x^2)_+^s $. From (a) to (d): $ s = \alpha, \, 2, \, 3, \, 4 $, where $ \alpha = 0.5 $ (blue), $ 1 $ (red), or $ 1.7 $ (green)
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Figure 8.  Comparison of our method with linear basis (i.e., $ \varphi^1 $) or quadratic basis (i.e., $ \varphi^2 $) and the finite difference method in [10] (i.e., FDM) in solving the fractional Poisson equation with exact solution $ u(x) = (1-x^2)^{4}_+ $, where $ \alpha = 0.5 $ (a) or $ 1.7 $ (b)
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Figure 9.  Numerical errors $ \|e_u\|_\infty $ in solving the 1D tempered fractional Poisson problem with exact solution $ u(x) = (1-x^2)_+^2 $, where $ \alpha = 0.6 $ (blue), $ 1 $ (red), or $ 1.5 $ (green)
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Figure 10.  Numerical errors in the solution of the 2D fractional Poisson problems with basis function $ \varphi^1 $ and mesh size $ h = 1/64 $
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Table 1.  Numerical errors $ \|e_ \Delta\|_\infty $ and convergence rates (c.r.) in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u $ with $ u = (1-x^2)_+^{1+\lfloor \alpha \rfloor} $ and basis function $ \varphi^p $ (for $ p = 0, 1, 2 $)
$ h $ 1/16 1/32 1/64 1/128 1/256 1/512
$ \alpha = 0.5 $
$ \varphi^0 $ 7.5879e-3 5.3220e-3 3.7499e-3 2.6475e-3 1.8707e-3 1.3224e-3
c.r. 0.5117 0.5051 0.5023 0.5010 0.5005
$ \varphi^1 $ 7.8596e-3 5.4986e-3 3.8696e-3 2.7304e-3 1.9287e-3 1.3632e-3
c.r. 0.5154 0.5069 0.5031 0.5014 0.5007
$ \varphi^2 $ 1.3338e-2 9.3477e-3 6.5851e-3 4.6488e-3 3.2848e-3 2.3219e-3
c.r. 0.5129 0.5054 0.5024 0.5011 0.5005
$ \alpha = 1 $
$ \varphi^0 $ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5
c.r. 1.0918 1.0197 1.0029 0.9997 0.9994
$ \varphi^1 $ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5
c.r. 1.0918 1.0197 1.0029 0.9997 0.9994
$ \varphi^2 $ 4.7698e-3 2.3572e-3 1.1717e-3 5.8413e-4 2.9164e-4 1.4571e-4
c.r. 1.0169 1.0085 1.0042 1.0021 1.0011
$ \alpha = 1.7 $
$ \varphi^0 $ 3.6356e-3 1.8041e-3 1.2777e-3 1.0195e-3 8.3276e-4 6.8083e-4
c.r. 1.0109 0.4977 0.3257 0.2919 0.2906
$ \varphi^1 $ 2.5288e-3 5.4873e-4 5.8948e-4 5.0041e-4 4.0137e-4 3.2097e-4
c.r. 2.2043 -0.1033 0.2363 0.3182 0.3225
$ \varphi^2 $ 9.9878e-2 7.8950e-2 6.3253e-2 5.1024e-2 4.1302e-2 3.3489e-2
c.r. 0.3392 0.3198 0.3010 0.3050 0.3025
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$ h $ 1/16 1/32 1/64 1/128 1/256 1/512
$ \alpha = 0.5 $
$ \varphi^0 $ 7.5879e-3 5.3220e-3 3.7499e-3 2.6475e-3 1.8707e-3 1.3224e-3
c.r. 0.5117 0.5051 0.5023 0.5010 0.5005
$ \varphi^1 $ 7.8596e-3 5.4986e-3 3.8696e-3 2.7304e-3 1.9287e-3 1.3632e-3
c.r. 0.5154 0.5069 0.5031 0.5014 0.5007
$ \varphi^2 $ 1.3338e-2 9.3477e-3 6.5851e-3 4.6488e-3 3.2848e-3 2.3219e-3
c.r. 0.5129 0.5054 0.5024 0.5011 0.5005
$ \alpha = 1 $
$ \varphi^0 $ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5
c.r. 1.0918 1.0197 1.0029 0.9997 0.9994
$ \varphi^1 $ 8.1722e-4 3.8342e-4 1.8911e-4 9.4360e-5 4.7189e-5 2.3604e-5
c.r. 1.0918 1.0197 1.0029 0.9997 0.9994
$ \varphi^2 $ 4.7698e-3 2.3572e-3 1.1717e-3 5.8413e-4 2.9164e-4 1.4571e-4
c.r. 1.0169 1.0085 1.0042 1.0021 1.0011
$ \alpha = 1.7 $
$ \varphi^0 $ 3.6356e-3 1.8041e-3 1.2777e-3 1.0195e-3 8.3276e-4 6.8083e-4
c.r. 1.0109 0.4977 0.3257 0.2919 0.2906
$ \varphi^1 $ 2.5288e-3 5.4873e-4 5.8948e-4 5.0041e-4 4.0137e-4 3.2097e-4
c.r. 2.2043 -0.1033 0.2363 0.3182 0.3225
$ \varphi^2 $ 9.9878e-2 7.8950e-2 6.3253e-2 5.1024e-2 4.1302e-2 3.3489e-2
c.r. 0.3392 0.3198 0.3010 0.3050 0.3025
Table 2.  Numerical errors $ \|e_ \Delta\|_\infty $ and convergence rate (c.r.) in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u $ with $ u = (1-x^2)_+^{2.1+ \alpha} $ and basis function $ \varphi^p $ (for $ p = 0, 1, 2 $).
$ h $ 1/16 1/32 1/64 1/128 1/256 1/512
$ \alpha = 0.5 $
$ \varphi^0 $ 9.7624e-5 2.8827e-5 7.6872e-6 1.9687e-6 4.9667e-7 1.2457e-7
c.r. 1.7598 1.9069 1.9653 1.9868 1.9953
$ \varphi^1 $ 2.1391e-4 5.9663e-5 1.5540e-5 3.9426e-6 9.9077e-7 2.4817e-7
c.r. 1.8421 1.9409 1.9787 1.9925 1.9972
$ \varphi^2 $ 1.0716e-4 2.4168e-5 5.5431e-6 1.2823e-6 2.9789e-7 6.9347e-8
c.r. 2.1486 2.1243 2.1120 2.1059 2.1029
$ \alpha = 1 $
$ \varphi^0 $ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7
c.r. 2.9767 2.9857 2.2109 1.6584 1.9189
$ \varphi^1 $ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7
c.r. 2.9767 2.9857 2.2109 1.6584 1.9189
$ \varphi^2 $ 2.4438e-4 5.4962e-5 1.2583e-5 2.9079e-6 6.7516e-7 1.5713e-7
c.r. 2.1526 2.1270 2.1134 2.1067 2.1033
$ \alpha = 1.7 $
$ \varphi^0 $ 1.4385e-2 5.3211e-3 1.5476e-3 4.0205e-4 9.9477e-5 2.4066e-5
c.r. 1.4348 1.7817 1.9446 2.0150 2.0474
$ \varphi^1 $ 1.5206e-2 5.5501e-3 1.6318e-3 4.2867e-4 1.0729e-4 2.6267e-5
c.r. 1.4540 1.7660 1.9285 1.9983 2.0302
$ \varphi^2 $ 8.0506e-4 1.0070e-4 1.6011e-5 3.1025e-6 6.6674e-7 1.4996e-7
c.r. 2.9991 2.6529 2.3675 2.2182 2.1526
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$ h $ 1/16 1/32 1/64 1/128 1/256 1/512
$ \alpha = 0.5 $
$ \varphi^0 $ 9.7624e-5 2.8827e-5 7.6872e-6 1.9687e-6 4.9667e-7 1.2457e-7
c.r. 1.7598 1.9069 1.9653 1.9868 1.9953
$ \varphi^1 $ 2.1391e-4 5.9663e-5 1.5540e-5 3.9426e-6 9.9077e-7 2.4817e-7
c.r. 1.8421 1.9409 1.9787 1.9925 1.9972
$ \varphi^2 $ 1.0716e-4 2.4168e-5 5.5431e-6 1.2823e-6 2.9789e-7 6.9347e-8
c.r. 2.1486 2.1243 2.1120 2.1059 2.1029
$ \alpha = 1 $
$ \varphi^0 $ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7
c.r. 2.9767 2.9857 2.2109 1.6584 1.9189
$ \varphi^1 $ 5.9137e-4 7.5126e-5 9.4842e-6 2.0487e-6 6.4898e-7 1.7163e-7
c.r. 2.9767 2.9857 2.2109 1.6584 1.9189
$ \varphi^2 $ 2.4438e-4 5.4962e-5 1.2583e-5 2.9079e-6 6.7516e-7 1.5713e-7
c.r. 2.1526 2.1270 2.1134 2.1067 2.1033
$ \alpha = 1.7 $
$ \varphi^0 $ 1.4385e-2 5.3211e-3 1.5476e-3 4.0205e-4 9.9477e-5 2.4066e-5
c.r. 1.4348 1.7817 1.9446 2.0150 2.0474
$ \varphi^1 $ 1.5206e-2 5.5501e-3 1.6318e-3 4.2867e-4 1.0729e-4 2.6267e-5
c.r. 1.4540 1.7660 1.9285 1.9983 2.0302
$ \varphi^2 $ 8.0506e-4 1.0070e-4 1.6011e-5 3.1025e-6 6.6674e-7 1.4996e-7
c.r. 2.9991 2.6529 2.3675 2.2182 2.1526
Table 3.  Numerical errors $ \|e_ \Delta\|_\infty $ and convergence rate (c.r.) in approximating function $ (- \Delta)^\frac{{ \alpha}}{{2}}u $ on (-1, 1) with $ u(x) = 1/(1+x^2) $ and basis function $ \varphi^p $ (for $ p = 0, 1, 2 $)
$ h $ 1/16 1/32 1/64 1/128 1/256
$ \alpha = 0.5 $
$ \varphi^0 $ 2.0023e-5 5.4222e-6 1.3919e-6 3.5121e-7 8.8084e-8
c.r. 1.8847 1.9618 1.9866 1.9954
$ \varphi^1 $ 3.0000e-5 8.3090e-6 2.1718e-6 5.5132e-7 1.3857e-7
c.r. 1.8522 1.9357 1.9780 1.9923
$ \varphi^2 $ 1.7384e-7 1.2148e-8 7.8796e-10 4.9569e-11 2.6986e-12
c.r. 3.8391 3.9464 3.9906 4.1991
$ \alpha = 1 $
$ \varphi^0 $ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7
c.r. 2.6193 2.4530 2.2945 2.1735
$ \varphi^1 $ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7
c.r. 2.6193 2.4530 2.2945 2.1735
$ \varphi^2 $ 8.0637e-7 2.5951e-8 8.3813e-10 2.7735e-11 9.3070e-13
c.r. 4.9576 4.9525 4.9174 4.8972
$ \alpha = 1.7 $
$ \varphi^0 $ 2.2284e-3 4.6838e-4 9.8834e-5 2.0988e-5 4.4908e-6
c.r. 2.2503 2.2446 2.2355 2.2245
$ \varphi^1 $ 2.3784e-3 5.1283e-4 1.1135e-4 2.4403e-5 5.4026e-6
c.r. 2.2135 2.2033 2.1900 2.1753
$ \varphi^2 $ 3.1706e-5 1.6865e-6 8.9337e-8 4.7546e-9 2.6046e-10
c.r. 4.2326 4.2386 4.2319 4.1902
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$ h $ 1/16 1/32 1/64 1/128 1/256
$ \alpha = 0.5 $
$ \varphi^0 $ 2.0023e-5 5.4222e-6 1.3919e-6 3.5121e-7 8.8084e-8
c.r. 1.8847 1.9618 1.9866 1.9954
$ \varphi^1 $ 3.0000e-5 8.3090e-6 2.1718e-6 5.5132e-7 1.3857e-7
c.r. 1.8522 1.9357 1.9780 1.9923
$ \varphi^2 $ 1.7384e-7 1.2148e-8 7.8796e-10 4.9569e-11 2.6986e-12
c.r. 3.8391 3.9464 3.9906 4.1991
$ \alpha = 1 $
$ \varphi^0 $ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7
c.r. 2.6193 2.4530 2.2945 2.1735
$ \varphi^1 $ 1.1056e-4 1.7994e-5 3.2863e-6 6.6985e-7 1.4849e-7
c.r. 2.6193 2.4530 2.2945 2.1735
$ \varphi^2 $ 8.0637e-7 2.5951e-8 8.3813e-10 2.7735e-11 9.3070e-13
c.r. 4.9576 4.9525 4.9174 4.8972
$ \alpha = 1.7 $
$ \varphi^0 $ 2.2284e-3 4.6838e-4 9.8834e-5 2.0988e-5 4.4908e-6
c.r. 2.2503 2.2446 2.2355 2.2245
$ \varphi^1 $ 2.3784e-3 5.1283e-4 1.1135e-4 2.4403e-5 5.4026e-6
c.r. 2.2135 2.2033 2.1900 2.1753
$ \varphi^2 $ 3.1706e-5 1.6865e-6 8.9337e-8 4.7546e-9 2.6046e-10
c.r. 4.2326 4.2386 4.2319 4.1902
Table 4.  Numerical errors $ \|e_u\|_\infty $ and convergence rate (c.r.) in solving the 1D Poisson problem on $ \Omega = (-1, 1) $, where $ f(x) = 1 $ in (1) and $ g(x) = 0 $ in (2)
$ h $ 1/16 1/32 1/64 1/128 1/256 1/512
$ \alpha = 0.6 $
$ \varphi^0 $ 7.4494e-2 5.9980e-2 4.8507e-2 3.9314e-2 3.1898e-2 2.5895e-2
c.r. 0.3126 0.3063 0.3031 0.3016 0.3008
$ \varphi^1 $ 7.5493e-2 6.0790e-2 4.9164e-2 3.9847e-2 3.2331e-2 2.6247e-2
c.r. 0.3125 0.3062 0.3031 0.3016 0.3008
$ \varphi^2 $ 8.4532e-2 6.8106e-2 5.5102e-2 4.4671e-2 3.6249e-2 2.9429e-2
c.r. 0.3117 0.3057 0.3028 0.3014 0.3007
$ \alpha = 1 $
$ \varphi^0 $ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3
c.r. 0.5107 0.5054 0.5027 0.5013 0.5007
$ \varphi^1 $ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3
c.r. 0.5107 0.5054 0.5027 0.5013 0.5007
$ \varphi^2 $ 5.7935e-2 4.0695e-2 2.8682e-2 2.0248e-2 1.4306e-2 1.0112e-2
c.r. 0.5096 0.5047 0.5023 0.5012 0.5006
$ \alpha = 1.5 $
$ \varphi^0 $ 1.6161e-2 9.5429e-3 5.6545e-3 3.3563e-3 1.9939e-3 1.1851e-3
c.r. 0.7600 0.7550 0.7525 0.7513 0.7506
$ \varphi^1 $ 1.5976e-2 9.4344e-3 5.5905e-3 3.3184e-3 1.9714e-3 1.1717e-3
c.r. 0.7599 0.7550 0.7525 0.7513 0.7506
$ \varphi^2 $ 2.2627e-2 1.3365e-2 7.9205e-3 4.7018e-3 2.7934e-3 1.6603e-3
c.r. 0.7596 0.7548 0.7524 0.7512 0.7506
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$ h $ 1/16 1/32 1/64 1/128 1/256 1/512
$ \alpha = 0.6 $
$ \varphi^0 $ 7.4494e-2 5.9980e-2 4.8507e-2 3.9314e-2 3.1898e-2 2.5895e-2
c.r. 0.3126 0.3063 0.3031 0.3016 0.3008
$ \varphi^1 $ 7.5493e-2 6.0790e-2 4.9164e-2 3.9847e-2 3.2331e-2 2.6247e-2
c.r. 0.3125 0.3062 0.3031 0.3016 0.3008
$ \varphi^2 $ 8.4532e-2 6.8106e-2 5.5102e-2 4.4671e-2 3.6249e-2 2.9429e-2
c.r. 0.3117 0.3057 0.3028 0.3014 0.3007
$ \alpha = 1 $
$ \varphi^0 $ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3
c.r. 0.5107 0.5054 0.5027 0.5013 0.5007
$ \varphi^1 $ 4.9166e-2 3.4508e-2 2.4310e-2 1.7158e-2 1.2121e-2 8.5671e-3
c.r. 0.5107 0.5054 0.5027 0.5013 0.5007
$ \varphi^2 $ 5.7935e-2 4.0695e-2 2.8682e-2 2.0248e-2 1.4306e-2 1.0112e-2
c.r. 0.5096 0.5047 0.5023 0.5012 0.5006
$ \alpha = 1.5 $
$ \varphi^0 $ 1.6161e-2 9.5429e-3 5.6545e-3 3.3563e-3 1.9939e-3 1.1851e-3
c.r. 0.7600 0.7550 0.7525 0.7513 0.7506
$ \varphi^1 $ 1.5976e-2 9.4344e-3 5.5905e-3 3.3184e-3 1.9714e-3 1.1717e-3
c.r. 0.7599 0.7550 0.7525 0.7513 0.7506
$ \varphi^2 $ 2.2627e-2 1.3365e-2 7.9205e-3 4.7018e-3 2.7934e-3 1.6603e-3
c.r. 0.7596 0.7548 0.7524 0.7512 0.7506
Table 5.  Numerical errors $ \|e_u\|_\infty $ and convergence rate (c.r.) in solving the 2D Poisson problem on $ \Omega = (-1, 1)^2 $ with $ f $ and $ g $ defined in (29)–(30), where linear basis $ \varphi^1 $ is used
$ h $ 1/4 1/8 1/16 1/32 1/64 1/128
$ \alpha = 0.2 $ 5.3181e-4 1.1946e-4 2.8883e-5 7.1505e-6 1.7827e-6 4.4531e-7
c.r. 2.1544 2.0482 2.0141 2.0040 2.0011
$ \alpha = 0.7 $ 2.3855e-3 5.1805e-4 1.2151e-4 2.9565e-5 7.3092e-6 1.8190e-6
c.r. 2.2031 2.0921 2.0391 2.0161 2.0065
$ \alpha = 1 $ 3.9406e-3 8.3747e-4 1.9083e-4 4.5384e-5 1.1056e-5 2.7276e-6
c.r. 2.2343 2.1338 2.0720 2.0374 2.0191
$ \alpha = 1.4 $ 6.9983e-3 1.4880e-3 3.2910e-4 7.5175e-5 1.7618e-5 4.2102e-6
c.r. 2.2337 2.1767 2.1302 2.0932 2.0651
$ \alpha = 1.9 $ 1.4264e-2 3.3824e-3 8.0943e-4 1.9424e-4 4.6676e-5 1.1230e-5
c.r. 2.0762 2.0631 2.0591 2.0571 2.0554
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$ h $ 1/4 1/8 1/16 1/32 1/64 1/128
$ \alpha = 0.2 $ 5.3181e-4 1.1946e-4 2.8883e-5 7.1505e-6 1.7827e-6 4.4531e-7
c.r. 2.1544 2.0482 2.0141 2.0040 2.0011
$ \alpha = 0.7 $ 2.3855e-3 5.1805e-4 1.2151e-4 2.9565e-5 7.3092e-6 1.8190e-6
c.r. 2.2031 2.0921 2.0391 2.0161 2.0065
$ \alpha = 1 $ 3.9406e-3 8.3747e-4 1.9083e-4 4.5384e-5 1.1056e-5 2.7276e-6
c.r. 2.2343 2.1338 2.0720 2.0374 2.0191
$ \alpha = 1.4 $ 6.9983e-3 1.4880e-3 3.2910e-4 7.5175e-5 1.7618e-5 4.2102e-6
c.r. 2.2337 2.1767 2.1302 2.0932 2.0651
$ \alpha = 1.9 $ 1.4264e-2 3.3824e-3 8.0943e-4 1.9424e-4 4.6676e-5 1.1230e-5
c.r. 2.0762 2.0631 2.0591 2.0571 2.0554
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