November  2017, 22(9): 3529-3545. doi: 10.3934/dcdsb.2017178

A preconditioned fast Hermite finite element method for space-fractional diffusion equations

1. 

School of Mathematics, Shandong University, Jinan, Shandong 250100, China

2. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

* Corresponding author: Aijie Cheng

Received  August 2016 Revised  April 2017 Published  July 2017

We develop a fast Hermite finite element method for a one-dimensional space-fractional diffusion equation, by proving that the stiffness matrix of the method can be expressed as a Toeplitz block matrix. Then a block circulant preconditioner is presented. Numerical results are presented to show the utility of the fast method.

Citation: Meng Zhao, Aijie Cheng, Hong Wang. A preconditioned fast Hermite finite element method for space-fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3529-3545. doi: 10.3934/dcdsb.2017178
References:
[1]

T. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Mod., 9 (2012), 658-666. 

[2]

D. BensonS. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413-1423.  doi: 10.1029/2000WR900032.

[3]

W. BuY. Tang and J. Yang, Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26-38.  doi: 10.1016/j.jcp.2014.07.023.

[4]

R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), 427-482.  doi: 10.1137/S0036144594276474.

[5]

R. H. Chan, Circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Matrix Analysis and Applications, 10 (1989), 542-550.  doi: 10.1137/0610039.

[6]

R. H. Chan and X. Q. Jin, A family of block preconditioners for block systems, SIAM journal on scientific and statistical computing, 13 (1992), 1218-1235.  doi: 10.1137/0913070.

[7]

P. J. Davis, Circulant Matrices, Wiley-Intersciences, New York, 1979.

[8]

W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47 (2008), 204-226.  doi: 10.1137/080714130.

[9]

K. Diethelm and A. D. Freed, An efficient algorithm for the evaluation of convolution integrals, Computers and Mathematics with Applications, 51 (2006), 204-226.  doi: 10.1016/j.camwa.2005.07.010.

[10]

N. Du and H. Wang, A fast finite element method for space-fractional dispersion equations on bounded domains in $\mathbb{R}^2$, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.  doi: 10.1137/15M1007458.

[11]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods for Partial Differential Equations, 22 (2005), 558-576.  doi: 10.1002/num.20112.

[12]

V. J. Ervin and J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in $R^{d}$, Numer. Methods for Partial Differential Equations, 23 (2007), 256-281.  doi: 10.1002/num.20169.

[13]

R. M. Gray, Toeplitz and circulant matrices: A review, Communications and Information Theory, 2 (2016), 155-239.  doi: 10.1561/0100000006.

[14]

J. Jia and H. Wang, A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842-862.  doi: 10.1016/j.jcp.2015.06.028.

[15]

Y. Jiang and X. Xu, Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), 374-392.  doi: 10.1016/j.jcp.2015.08.052.

[16]

S. L. Lei and H. W. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725.  doi: 10.1016/j.jcp.2013.02.025.

[17]

C. LiZ. Zhao and Y. Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers and Mathematics with Applications, 62 (2011), 855-875.  doi: 10.1016/j.camwa.2011.02.045.

[18]

X. Li and C. Xu, The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016-1051.  doi: 10.4208/cicp.020709.221209a.

[19]

F. R. LinS. W. Yang and X. Q. Jin, Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109-117.  doi: 10.1016/j.jcp.2013.07.040.

[20]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[21]

F. LiuV. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2014), 209-219.  doi: 10.1016/j.cam.2003.09.028.

[22]

C.~W. Lv and C. Xu, Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384-400. 

[23]

M. M. MeerschaertH. P. Scheffler and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249-261.  doi: 10.1016/j.jcp.2005.05.017.

[24]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.

[25]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[26]

H. K. Pang and H. W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.  doi: 10.1016/j.jcp.2011.10.005.

[27]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

[28]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. , SIAM, Rhode Island, 2003. doi: 10.1137/1. 9780898718003.

[29]

H. G. SunW. Chen and Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 389 (2010), 2719-2724.  doi: 10.1016/j.physa.2010.02.030.

[30]

H. TianH. Wang and W. Q. Wang, An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Mod., 10 (2013), 815-825. 

[31]

P. Vabishchevich, Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Mod., 13 (2016), 296-309. 

[32]

H. Wang and D. P. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.  doi: 10.1137/120892295.

[33]

H. WangD. P. Yang and S. F. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, Journal of Scientific Computing, 70 (2017), 429-449.  doi: 10.1007/s10915-016-0196-7.

[34]

H. Wang and X. H. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys., 281 (2015), 67-81.  doi: 10.1016/j.jcp.2014.10.018.

[35]

H. Wang and T. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458.  doi: 10.1137/12086491X.

[36]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49-57.  doi: 10.1016/j.jcp.2012.07.045.

[37]

H. WangK. Wang and T. Sircar, A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.  doi: 10.1016/j.jcp.2010.07.011.

[38]

L. L. WeiY. N. He and Y. Zhang, Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Mod., 10 (2013), 430-444. 

show all references

References:
[1]

T. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Mod., 9 (2012), 658-666. 

[2]

D. BensonS. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413-1423.  doi: 10.1029/2000WR900032.

[3]

W. BuY. Tang and J. Yang, Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26-38.  doi: 10.1016/j.jcp.2014.07.023.

[4]

R. H. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), 427-482.  doi: 10.1137/S0036144594276474.

[5]

R. H. Chan, Circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Matrix Analysis and Applications, 10 (1989), 542-550.  doi: 10.1137/0610039.

[6]

R. H. Chan and X. Q. Jin, A family of block preconditioners for block systems, SIAM journal on scientific and statistical computing, 13 (1992), 1218-1235.  doi: 10.1137/0913070.

[7]

P. J. Davis, Circulant Matrices, Wiley-Intersciences, New York, 1979.

[8]

W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47 (2008), 204-226.  doi: 10.1137/080714130.

[9]

K. Diethelm and A. D. Freed, An efficient algorithm for the evaluation of convolution integrals, Computers and Mathematics with Applications, 51 (2006), 204-226.  doi: 10.1016/j.camwa.2005.07.010.

[10]

N. Du and H. Wang, A fast finite element method for space-fractional dispersion equations on bounded domains in $\mathbb{R}^2$, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.  doi: 10.1137/15M1007458.

[11]

V. J. Ervin and J. P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods for Partial Differential Equations, 22 (2005), 558-576.  doi: 10.1002/num.20112.

[12]

V. J. Ervin and J. P. Roop, Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in $R^{d}$, Numer. Methods for Partial Differential Equations, 23 (2007), 256-281.  doi: 10.1002/num.20169.

[13]

R. M. Gray, Toeplitz and circulant matrices: A review, Communications and Information Theory, 2 (2016), 155-239.  doi: 10.1561/0100000006.

[14]

J. Jia and H. Wang, A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842-862.  doi: 10.1016/j.jcp.2015.06.028.

[15]

Y. Jiang and X. Xu, Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), 374-392.  doi: 10.1016/j.jcp.2015.08.052.

[16]

S. L. Lei and H. W. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725.  doi: 10.1016/j.jcp.2013.02.025.

[17]

C. LiZ. Zhao and Y. Q. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers and Mathematics with Applications, 62 (2011), 855-875.  doi: 10.1016/j.camwa.2011.02.045.

[18]

X. Li and C. Xu, The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016-1051.  doi: 10.4208/cicp.020709.221209a.

[19]

F. R. LinS. W. Yang and X. Q. Jin, Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109-117.  doi: 10.1016/j.jcp.2013.07.040.

[20]

Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.  doi: 10.1016/j.jcp.2007.02.001.

[21]

F. LiuV. Anh and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2014), 209-219.  doi: 10.1016/j.cam.2003.09.028.

[22]

C.~W. Lv and C. Xu, Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384-400. 

[23]

M. M. MeerschaertH. P. Scheffler and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249-261.  doi: 10.1016/j.jcp.2005.05.017.

[24]

M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.  doi: 10.1016/j.cam.2004.01.033.

[25]

R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3.

[26]

H. K. Pang and H. W. Sun, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.  doi: 10.1016/j.jcp.2011.10.005.

[27]

I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

[28]

Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. , SIAM, Rhode Island, 2003. doi: 10.1137/1. 9780898718003.

[29]

H. G. SunW. Chen and Y. Q. Chen, Fractional differential models for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 389 (2010), 2719-2724.  doi: 10.1016/j.physa.2010.02.030.

[30]

H. TianH. Wang and W. Q. Wang, An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Mod., 10 (2013), 815-825. 

[31]

P. Vabishchevich, Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Mod., 13 (2016), 296-309. 

[32]

H. Wang and D. P. Yang, Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.  doi: 10.1137/120892295.

[33]

H. WangD. P. Yang and S. F. Zhu, Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, Journal of Scientific Computing, 70 (2017), 429-449.  doi: 10.1007/s10915-016-0196-7.

[34]

H. Wang and X. H. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys., 281 (2015), 67-81.  doi: 10.1016/j.jcp.2014.10.018.

[35]

H. Wang and T. Basu, A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458.  doi: 10.1137/12086491X.

[36]

H. Wang and N. Du, A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49-57.  doi: 10.1016/j.jcp.2012.07.045.

[37]

H. WangK. Wang and T. Sircar, A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.  doi: 10.1016/j.jcp.2010.07.011.

[38]

L. L. WeiY. N. He and Y. Zhang, Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Mod., 10 (2013), 430-444. 

Table 1.  The condition number of stiffness matrix $\mathsf{A}$ with $K=1$, $\gamma=0.5$, $\beta=0.5$
h $2^{-8} $ $2^{-9} $$2^{-10} $
$Cond(\mathsf{A})$ $ 2.329917 \times 10^{6}$ $ 9.320396 \times 10^{6}$ $ 3.728232 \times 10^{7}$
h $2^{-8} $ $2^{-9} $$2^{-10} $
$Cond(\mathsf{A})$ $ 2.329917 \times 10^{6}$ $ 9.320396 \times 10^{6}$ $ 3.728232 \times 10^{7}$
Table 2.  The $L^2$ error and $H^{1-\frac{\beta}{2}}$ error of the cubic Hermite element method with $\beta=0.1,0.5,0.9$
$\beta$hDOF $ \|u_h - u \|_{L^2}$order $ \|u_h - u \|_{H^{1-\frac{\beta}{2}}}$order
0.5 $2^{-3} $16 $ 4.872892 \times 10^{-4}$ $ 2.512934 \times 10^{-3}$
$2^{-4} $32 $ 3.573631 \times 10^{-5}$3.7693 $ 2.576521 \times 10^{-4}$3.2858
$2^{-5} $64 $ 2.236217 \times 10^{-6}$3.9982 $ 2.481970 \times 10^{-5}$3.3758
$2^{-6} $128$ 1.342295 \times 10^{-7}$4.0582$ 2.249708 \times 10^{-6}$3.4636
$2^{-7} $256$ 7.909661 \times 10^{-9}$4.0849$ 2.033970 \times 10^{-7}$3.4673
0.1 $2^{-3} $16 $ 7.851432 \times 10^{-4}$ $ 8.011606 \times 10^{-3}$
$2^{-4} $32 $ 6.192597 \times 10^{-5}$3.6643 $ 1.304843 \times 10^{-3}$2.6352
$2^{-5} $64 $ 4.208159 \times 10^{-6}$3.8792 $ 1.805192 \times 10^{-4}$2.8536
$2^{-6} $128$ 2.713900 \times 10^{-7}$3.9547$ 2.295487 \times 10^{-5}$2.9752
$2^{-7} $256$ 1.794745 \times 10^{-8}$3.9185$ 2.808095 \times 10^{-6}$3.0311
0.9 $2^{-3} $16 $ 3.622149 \times 10^{-4}$ $ 4.846137 \times 10^{-4}$
$2^{-4} $32 $ 2.767086 \times 10^{-5}$3.7104 $ 4.110739 \times 10^{-5}$3.5593
$2^{-5} $64 $ 1.826287 \times 10^{-6}$3.9213 $ 3.012015 \times 10^{-6}$3.7705
$2^{-6} $128$ 1.180438 \times 10^{-7}$3.9515$ 2.132241 \times 10^{-7}$3.8202
$2^{-7} $256$ 7.376961 \times 10^{-9}$4.0001$ 1.415327 \times 10^{-8}$3.9131
$\beta$hDOF $ \|u_h - u \|_{L^2}$order $ \|u_h - u \|_{H^{1-\frac{\beta}{2}}}$order
0.5 $2^{-3} $16 $ 4.872892 \times 10^{-4}$ $ 2.512934 \times 10^{-3}$
$2^{-4} $32 $ 3.573631 \times 10^{-5}$3.7693 $ 2.576521 \times 10^{-4}$3.2858
$2^{-5} $64 $ 2.236217 \times 10^{-6}$3.9982 $ 2.481970 \times 10^{-5}$3.3758
$2^{-6} $128$ 1.342295 \times 10^{-7}$4.0582$ 2.249708 \times 10^{-6}$3.4636
$2^{-7} $256$ 7.909661 \times 10^{-9}$4.0849$ 2.033970 \times 10^{-7}$3.4673
0.1 $2^{-3} $16 $ 7.851432 \times 10^{-4}$ $ 8.011606 \times 10^{-3}$
$2^{-4} $32 $ 6.192597 \times 10^{-5}$3.6643 $ 1.304843 \times 10^{-3}$2.6352
$2^{-5} $64 $ 4.208159 \times 10^{-6}$3.8792 $ 1.805192 \times 10^{-4}$2.8536
$2^{-6} $128$ 2.713900 \times 10^{-7}$3.9547$ 2.295487 \times 10^{-5}$2.9752
$2^{-7} $256$ 1.794745 \times 10^{-8}$3.9185$ 2.808095 \times 10^{-6}$3.0311
0.9 $2^{-3} $16 $ 3.622149 \times 10^{-4}$ $ 4.846137 \times 10^{-4}$
$2^{-4} $32 $ 2.767086 \times 10^{-5}$3.7104 $ 4.110739 \times 10^{-5}$3.5593
$2^{-5} $64 $ 1.826287 \times 10^{-6}$3.9213 $ 3.012015 \times 10^{-6}$3.7705
$2^{-6} $128$ 1.180438 \times 10^{-7}$3.9515$ 2.132241 \times 10^{-7}$3.8202
$2^{-7} $256$ 7.376961 \times 10^{-9}$4.0001$ 1.415327 \times 10^{-8}$3.9131
Table 3.  Performance of Gauss, CGS methods with $\beta=0.5$
GaussCGS
hCPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.09160.2103137
$2^{-7}$0.48651.5369272
$2^{-8}$3.58589.8734415
$2^{-9}$24.477462.7682665
$2^{-10}$186.9874391.5595899
$2^{-11}$1485.14502731.07511676
$2^{-12}$out of memoryout of memory
GaussCGS
hCPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.09160.2103137
$2^{-7}$0.48651.5369272
$2^{-8}$3.58589.8734415
$2^{-9}$24.477462.7682665
$2^{-10}$186.9874391.5595899
$2^{-11}$1485.14502731.07511676
$2^{-12}$out of memoryout of memory
Table 4.  Performance of FCGS, SFCGS and PFCGS methods with $\beta=0.5$
FCGSSFCGSCFCGS
hCPU(s)Itr. # CPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.08321370.030570.03419
$2^{-7}$0.18092720.035870.041910
$2^{-8}$0.28904150.050370.044511
$2^{-9}$0.99076650.068080.065312
$2^{-10}$2.45258990.093280.104614
$2^{-11}$16.075216760.153690.242216
$2^{-12}$26.475164210.191490.595519
$2^{-13}$N/A $>$ 30,0000.6319101.410722
FCGSSFCGSCFCGS
hCPU(s)Itr. # CPU(s)Itr. # CPU(s)Itr. #
$2^{-6}$0.08321370.030570.03419
$2^{-7}$0.18092720.035870.041910
$2^{-8}$0.28904150.050370.044511
$2^{-9}$0.99076650.068080.065312
$2^{-10}$2.45258990.093280.104614
$2^{-11}$16.075216760.153690.242216
$2^{-12}$26.475164210.191490.595519
$2^{-13}$N/A $>$ 30,0000.6319101.410722
[1]

Hong Wang, Aijie Cheng, Kaixin Wang. Fast finite volume methods for space-fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1427-1441. doi: 10.3934/dcdsb.2015.20.1427

[2]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402

[3]

Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems and Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195

[4]

Xing Li, Chungen Shen, Lei-Hong Zhang. A projected preconditioned conjugate gradient method for the linear response eigenvalue problem. Numerical Algebra, Control and Optimization, 2018, 8 (4) : 389-412. doi: 10.3934/naco.2018025

[5]

Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic and Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59

[6]

Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018

[7]

El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883

[8]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073

[9]

Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

[10]

Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085

[11]

Moulay Rchid Sidi Ammi, Ismail Jamiai. Finite difference and Legendre spectral method for a time-fractional diffusion-convection equation for image restoration. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 103-117. doi: 10.3934/dcdss.2018007

[12]

Shu-Yu Hsu. Super fast vanishing solutions of the fast diffusion equation. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5383-5414. doi: 10.3934/dcds.2020232

[13]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711

[14]

Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic and Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019

[15]

Junxiong Jia, Jigen Peng, Jinghuai Gao, Yujiao Li. Backward problem for a time-space fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (3) : 773-799. doi: 10.3934/ipi.2018033

[16]

Zhiguang Zhang, Qiang Liu, Tianling Gao. A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising. Inverse Problems and Imaging, 2021, 15 (6) : 1451-1469. doi: 10.3934/ipi.2021018

[17]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435

[18]

Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317

[19]

H. T. Liu. Impulsive effects on the existence of solutions for a fast diffusion equation. Conference Publications, 2001, 2001 (Special) : 248-253. doi: 10.3934/proc.2001.2001.248

[20]

Marek Fila, Juan-Luis Vázquez, Michael Winkler. A continuum of extinction rates for the fast diffusion equation. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1129-1147. doi: 10.3934/cpaa.2011.10.1129

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (293)
  • HTML views (129)
  • Cited by (5)

Other articles
by authors

[Back to Top]