Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion

We consider the initial boundary value problem for the inhomogeneous time-fractional diu- sion equation with a homogeneous Dirichlet boundary condition and a nonsmooth right hand side data in a bounded convex polyhedral domain. We analyze two semidiscrete schemes based on the standard Galerkin and lumped mass nite element methods. Almost optimal error estimates are obtained for right hand side data f(x; t)2 L 1 (0; T ; _ H q ()), 1 < q 1, for both semidiscrete schemes. For lumped mass method, the optimal L 2 ()-norm error estimate requires symmetric meshes. Finally, numerical experiments for one- and two-dimensional examples are presented to verify our theoretical results.

[1]  Xianjuan Li,et al.  A Space-Time Spectral Method for the Time Fractional Diffusion Equation , 2009, SIAM J. Numer. Anal..

[2]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[3]  V. Thomée Galerkin Finite Element Methods for Parabolic Problems (Springer Series in Computational Mathematics) , 2010 .

[4]  R. Nigmatullin The Realization of the Generalized Transfer Equation in a Medium with Fractal Geometry , 1986, January 1.

[5]  V. Thomée,et al.  Numerical solution via Laplace transforms of a fractional order evolution equation , 2010 .

[6]  V. Thomée,et al.  Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation , 2010 .

[7]  William McLean,et al.  Superconvergence of a Discontinuous Galerkin Method for Fractional Diffusion and Wave Equations , 2012, SIAM J. Numer. Anal..

[8]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[9]  L. Gelhar,et al.  Field study of dispersion in a heterogeneous aquifer: 2. Spatial moments analysis , 1992 .

[10]  G. Burton Sobolev Spaces , 2013 .

[11]  Jun Zou,et al.  Numerical Reconstruction of Heat Fluxes , 2005, SIAM J. Numer. Anal..

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

[13]  William McLean,et al.  Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation , 2009, Numerical Algorithms.

[14]  E. Montroll Random walks on lattices , 1969 .

[15]  Raytcho D. Lazarov,et al.  Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations , 2012, SIAM J. Numer. Anal..

[16]  Jun Zou,et al.  Numerical identifications of parameters in parabolic systems , 1998 .

[17]  Rudolf Hilfer,et al.  Numerical Algorithm for Calculating the Generalized Mittag-Leffler Function , 2008, SIAM J. Numer. Anal..

[18]  Naomichi Hatano,et al.  Dispersive transport of ions in column experiments: An explanation of long‐tailed profiles , 1998 .

[19]  Santos B. Yuste,et al.  An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations , 2004, SIAM J. Numer. Anal..

[20]  William McLean,et al.  Piecewise-linear, discontinuous Galerkin method for a fractional diffusion equation , 2011, Numerical Algorithms.

[21]  Chuanju Xu,et al.  Finite difference/spectral approximations for the time-fractional diffusion equation , 2007, J. Comput. Phys..

[22]  Massimiliano Giona,et al.  Fractional diffusion equation and relaxation in complex viscoelastic materials , 1992 .

[23]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[24]  Ralf Metzler,et al.  Physical pictures of transport in heterogeneous media: Advection‐dispersion, random‐walk, and fractional derivative formulations , 2002, cond-mat/0202327.

[25]  K. Mustapha An implicit finite-difference time-stepping method for a sub-diffusion equation, with spatial discretization by finite elements , 2011 .

[26]  Bangti Jin,et al.  Galerkin FEM for Fractional Order Parabolic Equations with Initial Data in H - s , 0 ≤ s ≤ 1 , 2012, NAA.

[27]  Masahiro Yamamoto,et al.  Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation , 2009 .

[28]  Vidar Thomée,et al.  The lumped mass finite element method for a parabolic problem , 1985, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.