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

Yixuan Wu; Yanzhi Zhang

doi:10.3934/dcdss.2022016

Article Contents

2022, Volume 15, Issue 4: 851-876. Doi: 10.3934/dcdss.2022016

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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
^* Corresponding author: Yanzhi Zhang

Received: February 28, 2021

Revised: November 30, 2021

Early access: February 2022

Published: April 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 65R20, 65N35, 45E05; Secondary: 47G10, 47G30.

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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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	c.r.	0.3392	0.3198	0.3010	0.3050	0.3025

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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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	c.r.	2.9991	2.6529	2.3675	2.2182	2.1526

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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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	c.r.	4.2326	4.2386	4.2319	4.1902

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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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^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
$ \varphi^0 $	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
$ \varphi^1 $	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
$ \varphi^2 $	c.r.	0.7596	0.7548	0.7524	0.7512	0.7506

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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
$ \alpha = 0.2 $	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
$ \alpha = 0.7 $	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
$ \alpha = 1 $	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
$ \alpha = 1.4 $	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
$ \alpha = 1.9 $	c.r.	2.0762	2.0631	2.0591	2.0571	2.0554

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