Error estimate for indirect spectral approximation of optimal control problem governed by fractional diffusion equation with variable diffusivity coefficient

Abstract In this paper an indirect spectral method of an optimal control problem governed by a space-fractional diffusion equation with variable diffusivity coefficient is studied. First-order optimality conditions of the proposed model are derived and the regularity of the solutions is analyzed. Indirect spectral methods via weighted Jacobi polynomials are built up based on the “first optimize, then discretize” strategy, and a priori error estimates of the discrete optimal control problem in weighted norms are derived. As proposed indirect spectral discrete schemes are designed to accommodate the impact of the variable coefficients, which are complicated and not formulated under the variational framework, conventional error estimate techniques of the discrete space-fractional optimal control problems do not apply. Novel treatments of the discrete variational inequality are developed to resolve the aforementioned issues and to support the error estimates. Numerical examples are presented to verify the theoretical findings.

[1]  V. Ervin,et al.  Existence and Regularity of solutions to 1-D Fractional Order Diffusion Equations , 2018, 1808.10555.

[2]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

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

[4]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[5]  Ivo Babuska,et al.  Direct and Inverse Approximation Theorems for the p-Version of the Finite Element Method in the Framework of Weighted Besov Spaces. Part I: Approximability of Functions in the Weighted Besov Spaces , 2001, SIAM J. Numer. Anal..

[6]  Shengyue Li,et al.  Fractional spectral collocation method for optimal control problem governed by space fractional diffusion equation , 2019, Appl. Math. Comput..

[7]  Delfim F. M. Torres,et al.  A discrete method to solve fractional optimal control problems , 2014, 1403.5060.

[8]  Zhaojie Zhou,et al.  Spectral Galerkin approximation of optimal control problem governed by Riesz fractional differential equation , 2019, Applied Numerical Mathematics.

[9]  O. Agrawal.  A General Formulation and Solution Scheme for Fractional Optimal Control Problems , 2004 .

[10]  Fawang Liu,et al.  Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation , 2007, Appl. Math. Comput..

[11]  Zhi-Zhong Sun,et al.  A fourth-order approximation of fractional derivatives with its applications , 2015, J. Comput. Phys..

[12]  Bangti Jin,et al.  Pointwise-in-time error estimates for an optimal control problem with subdiffusion constraint , 2017, IMA Journal of Numerical Analysis.

[13]  Cui-Cui Ji,et al.  Numerical Schemes for Solving the Time-Fractional Dual-Phase-Lagging Heat Conduction Model in a Double-Layered Nanoscale Thin Film , 2019, Journal of Scientific Computing.

[14]  J. Klafter,et al.  Anomalous diffusion spreads its wings , 2005 .

[15]  Hong Wang,et al.  A Fast Gradient Projection Method for a Constrained Fractional Optimal Control , 2016, J. Sci. Comput..

[16]  Chuanju Xu,et al.  Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem , 2013 .

[17]  Lueling Jia,et al.  Mixed-Type Galerkin Variational Principle and Numerical Simulation for a Generalized Nonlocal Elastic Model , 2017, J. Sci. Comput..

[18]  J. A. Tenreiro Machado,et al.  On the formulation and numerical simulation of distributed-order fractional optimal control problems , 2017, Commun. Nonlinear Sci. Numer. Simul..

[19]  Wei Gong,et al.  Finite element approximation of optimal control problems governed by time fractional diffusion equation , 2016, Comput. Math. Appl..

[20]  Zhiyu Tan,et al.  Finite Element Approximation of Optimal Control Problem Governed by Space Fractional Equation , 2018, J. Sci. Comput..

[21]  Xiangcheng Zheng,et al.  Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation , 2018, Applied Numerical Mathematics.

[22]  Zhongqiang Zhang,et al.  Optimal Regularity and Error Estimates of a Spectral Galerkin Method for Fractional Advection-Diffusion-Reaction Equations , 2020, SIAM J. Numer. Anal..

[23]  Xiangcheng Zheng,et al.  Numerical Approximations for the Variable Coefficient Fractional Diffusion Equations with Non-smooth Data , 2019, Comput. Methods Appl. Math..

[24]  Xiaoping Xie,et al.  Analysis of a Time-Stepping Scheme for Time Fractional Diffusion Problems with Nonsmooth Data , 2018, SIAM J. Numer. Anal..

[25]  Hai-Wei Sun,et al.  Fourth order finite difference schemes for time-space fractional sub-diffusion equations , 2016, Comput. Math. Appl..

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

[27]  Minghua Chen,et al.  High Order Algorithms for the Fractional Substantial Diffusion Equation with Truncated Lévy Flights , 2014, SIAM J. Sci. Comput..

[28]  Weihua Deng,et al.  A Hybridized Discontinuous Galerkin Method for 2D Fractional Convection–Diffusion Equations , 2015, Journal of Scientific Computing.

[29]  William Rundell,et al.  Variational formulation of problems involving fractional order differential operators , 2013, Math. Comput..

[30]  M. Zaky.  A Legendre collocation method for distributed-order fractional optimal control problems , 2018 .

[31]  D. Benson,et al.  The fractional‐order governing equation of Lévy Motion , 2000 .

[32]  Hong Wang,et al.  Wellposedness of Variable-Coefficient Conservative Fractional Elliptic Differential Equations , 2013, SIAM J. Numer. Anal..

[33]  Xinlong Feng,et al.  Investigations on several high-order ADI methods for time-space fractional diffusion equation , 2018, Numerical Algorithms.

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

[35]  Enrique Otárola,et al.  A Priori Error Estimates for the Optimal Control of the Integral Fractional Laplacian , 2019, SIAM J. Control. Optim..

[36]  Harbir Antil,et al.  A FEM for an Optimal Control Problem of Fractional Powers of Elliptic Operators , 2014, SIAM J. Control. Optim..

[37]  Hong Wang,et al.  A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations , 2013, J. Comput. Phys..

[38]  Nianyu Yi,et al.  A Legendre-Galerkin Spectral Method for Optimal Control Problems Governed by Elliptic Equations , 2008, SIAM J. Numer. Anal..

[39]  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..

[40]  Fenglin Huang,et al.  A Legendre-Galerkin Spectral Method for Optimal Control Problems Governed by Stokes Equations , 2011, SIAM J. Numer. Anal..