The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions. Nearly optimal error estimates for both cases of initial data and inhomogeneous term are derived, which cover both smooth and nonsmooth data. Further we develop a fully discrete scheme based on a finite difference discretization of the time-fractional derivatives, and discuss its stability and error estimate. Extensive numerical experiments for one and two-dimension problems confirm the convergence rates of the theoretical results.Comment: 22 pages, 4 figure

Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion

We consider the initial boundary value problem for the inhomogeneous time-fractional diffusion 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 finite element methods. Almost optimal error estimates are obtained for right hand side data $f(x,t)\in L^\infty(0,T;\dot H^q(\Omega))$, $-1< q \le 1$, for both semidiscrete schemes. For lumped mass method, the optimal $L^2(\Omega)$-norm error estimate requires symmetric meshes. Finally, numerical experiments for one- and two-dimensional examples are presented to verify our theoretical results.Comment: 21 pages, 4 figure

Numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation

In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the $H^1$-norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm.Comment: 17 pages, 2 figures, 2 table

Convergence analysis of a Crank-Nicolson Galerkin method for an inverse source problem for parabolic equations with boundary observations

This work is devoted to an inverse problem of identifying a source term depending on both spatial and time variables in a parabolic equation from single Cauchy data on a part of the boundary. A Crank-Nicolson Galerkin method is applied to the least squares functional with an quadratic stabilizing penalty term. The convergence of finite dimensional regularized approximations to the sought source as measurement noise levels and mesh sizes approach to zero with an appropriate regularization parameter is proved. Moreover, under a suitable source condition, an error bound and corresponding convergence rates are proved. Finally, several numerical experiments are presented to illustrate the theoretical findings.Comment: Inverse source problem, Tikhonov regularization, Crank-Nicolson Galerkin method, Source condition, Convergence rates, Ill-posedness, Parabolic proble

NUMERICAL RECONSTRUCTION OF HEAT FLUXES ∗

Abstract. This paper studies the reconstruction of heat fluxes on an inner boundary of a heat conductive system when the measurement of temperature in a small subregion near the outer boundary of the physical domain is available. We will first consider two different regularization formulations for this severely ill-posed inverse problem and justify their well-posedness; then we will propose two fully discrete finite element methods to approximate the resultant nonlinear minimization problems. The existence and uniqueness of the discrete minimizers and convergence of the finite element solution are rigorously demonstrated. A conjugate gradient method is formulated to solve the nonlinear finite element optimization problems. Numerical experiments are given to demonstrate the stability and effectiveness of the proposed reconstruction methods