A preconditioned fast finite element approximation to variable-order time-fractional diffusion equations in multiple space dimensions

Abstract We develop a preconditioned fast divided-and-conquer finite element approximation for the initial-boundary value problem of variable-order time-fractional diffusion equations. Due to the impact of the time-dependent variable order, the coefficient matrix of the resulting all-at-once system does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires O ( M N log 3 ⁡ N ) computational complexity and O ( M N log 2 ⁡ N ) memory with M and N being the numbers of degrees of freedom in space and time, respectively. Furthermore, a preconditioner is introduced to reduce the number of iterations caused by the bad condition number of the coefficient matrix. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

[1]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[2]  Zhimin Zhang,et al.  Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations , 2015, 1511.03453.

[3]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[4]  Zhongqiang Zhang,et al.  A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations , 2015, SIAM J. Sci. Comput..

[5]  Rina Schumer,et al.  Fractal mobile/immobile solute transport , 2003 .

[6]  Hongfei Fu,et al.  A preconditioned fast finite difference method for space-time fractional partial differential equations , 2017 .

[7]  Jinhong Jia,et al.  A fast method for variable-order space-fractional diffusion equations , 2019, Numer. Algorithms.

[8]  Jose L. Gracia,et al.  Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation , 2017, SIAM J. Numer. Anal..

[9]  George E. Karniadakis,et al.  Fractional Spectral Collocation Method , 2014, SIAM J. Sci. Comput..

[10]  Hong Wang,et al.  A divide-and-conquer fast finite difference method for space-time fractional partial differential equation , 2017, Comput. Math. Appl..

[11]  Jinhong Jia,et al.  A fast collocation approximation to a two-sided variable-order space-fractional diffusion equation and its analysis , 2021, J. Comput. Appl. Math..

[12]  Hongguang Sun,et al.  Finite difference Schemes for Variable-Order Time fractional Diffusion equation , 2012, Int. J. Bifurc. Chaos.

[13]  Rina Schumer,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[14]  Hong Wang,et al.  A direct O(N log2 N) finite difference method for fractional diffusion equations , 2010, J. Comput. Phys..

[15]  Feng Wang,et al.  Finite element simulation and efficient algorithm for fractional Cahn-Hilliard equation , 2019, J. Comput. Appl. Math..

[16]  Jinhong Jia,et al.  A fast finite volume method for conservative space-time fractional diffusion equations discretized on space-time locally refined meshes , 2019, Comput. Math. Appl..

[17]  Mihály Kovács,et al.  Reprint of: Boundary conditions for fractional diffusion , 2018, J. Comput. Appl. Math..

[18]  D. Benson,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[19]  Daniele Bertaccini,et al.  Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation , 2019, Appl. Math. Lett..

[20]  Norbert Heuer,et al.  Regularity of the solution to 1-D fractional order diffusion equations , 2016, Math. Comput..

[21]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[22]  Weihua Deng,et al.  Finite Element Method for the Space and Time Fractional Fokker-Planck Equation , 2008, SIAM J. Numer. Anal..

[23]  Michael K. Ng,et al.  A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations , 2015, J. Comput. Phys..

[24]  Xiao-Qing Jin,et al.  Numerical Linear Algebra and its Applications , 2004 .

[25]  Zhi‐zhong Sun,et al.  Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations: A Second-Order Scheme , 2017 .

[26]  Masahiro Yamamoto,et al.  Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems , 2011 .

[27]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[28]  Xuan Zhao,et al.  Second-order approximations for variable order fractional derivatives: Algorithms and applications , 2015, J. Comput. Phys..

[29]  Kai Diethelm,et al.  A note on the well-posedness of terminal value problems for fractional differential equations , 2018, Journal of Integral Equations and Applications.

[30]  Xiangcheng Zheng,et al.  Wellposedness and regularity of the variable-order time-fractional diffusion equations , 2019, Journal of Mathematical Analysis and Applications.

[31]  Xiangcheng Zheng,et al.  An Error Estimate of a Numerical Approximation to a Hidden-Memory Variable-Order Space-Time Fractional Diffusion Equation , 2020, SIAM J. Numer. Anal..

[32]  Igor Moret,et al.  Solving the time-fractional Schrödinger equation by Krylov projection methods , 2015, J. Comput. Phys..

[33]  Zhiping Mao,et al.  Efficient spectral-Galerkin methods for fractional partial differential equations with variable coefficients , 2016, J. Comput. Phys..

[34]  Mark M. Meerschaert,et al.  Limit theorem for continuous-time random walks with two time scales , 2004, Journal of Applied Probability.

[35]  Hong Wang,et al.  Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation , 2016, J. Comput. Appl. Math..

[36]  Fawang Liu,et al.  Numerical techniques for the variable order time fractional diffusion equation , 2012, Appl. Math. Comput..

[37]  Diane M. McKnight,et al.  Transport and cycling of iron and hydrogen peroxide in a freshwater stream: Influence of organic acids , 2003 .

[38]  Changpin Li,et al.  High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II) , 2015 .

[39]  Hongguang Sun,et al.  A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications , 2019, Fractional Calculus and Applied Analysis.

[40]  M. Meerschaert,et al.  Stochastic Models for Fractional Calculus , 2011 .

[41]  A Fast Finite Volume Method on Locally Refined Meshes for Fractional Diffusion Equations , 2019, East Asian Journal on Applied Mathematics.

[42]  Xiao-Qing Jin,et al.  Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations , 2015 .

[43]  Natalia Kopteva,et al.  Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions , 2017, Math. Comput..

[44]  Zhi-Zhong Sun,et al.  The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain , 2013, J. Comput. Phys..