This article concerns second-order time discretization of subdiffusion equations with time-dependent diffusion coefficients. High-order differentiability and regularity estimates are established for subdiffusion equations with time-dependent coefficients. Using these regularity results and a perturbation argument of freezing the diffusion coefficient, we prove that the convolution quadrature generated by the second-order backward differentiation formula, with proper correction at the first time step, can achieve second-order convergence for both nonsmooth initial data and incompatible source term. Numerical experiments are consistent with the theoretical results.
In 1986, Dixon and McKee developed a discrete fractional Grönwall inequality [Z. Angew. Math. Mech., 66 (1986), pp. 535--544], which can be seen as a generalization of the classical discrete Grönwall inequality. However, this generalized discrete Grönwall inequality has not been widely applied in the numerical analysis of the time-stepping methods for the time-fractional evolution equations. The main purpose of this paper is to show how to apply the generalized discrete Grönwall inequality to prove the convergence of a class of time-stepping numerical methods for time-fractional nonlinear subd…
The aim of this paper is to develop and analyze high-order time stepping schemes for solving semilinear subdiffusion equations. We apply the $k$-step BDF convolution quadrature to discretize the time-fractional derivative with order $α\in (0,1)$, and modify the starting steps in order to achieve optimal convergence rate. This method has already been well-studied for the linear fractional evolution equations in Jin, Li and Zhou \cite{JinLiZhou:correction}, while the numerical analysis for the nonlinear problem is still missing in the literature. By splitting the nonlinear potential term into an…
The aim of this paper is to study the time stepping scheme for approximately solving the subdiffusion equation with a weakly singular source term. In this case, many popular time stepping schemes, including the correction of high-order BDF methods, may lose their high-order accuracy. To fill in this gap, in this paper, we develop a novel time stepping scheme, where the source term is regularized by using a $k$-fold integral-derivative and the equation is discretized by using a modified BDF2 convolution quadrature. We prove that the proposed time stepping scheme is second-order, even if the sou…
Sharp $H^1$-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems
Henan University of Economics and Law（河南财经政法大学）；Nanjing University of Aeronautics and Astronautics（南京航空航天大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
Regularity and high-order time stepping for semilinear subdiffusion equations with singular initial data beyond the $L^\infty$ framework
Hunan Research Center of the Basic Discipline Fundamental Algorithmic Theory and Novel Computational Methods, School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院基础学科基本算法理论与新型计算方法湖南省研究中心）
An essential feature of the subdiffusion equations with the $α$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $σ\in (0,1)\cup(1,2)$. Under this general regularity assumption, we here obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. To the end, we present a refined discrete fractional-type Grönwall inequality and a rigorous analysis for the truncation errors. Numerical experiments are provided to demonstrate the effe…
This paper is devoted to the numerical analysis of a piecewise constant discontinuous Galerkin method for time fractional subdiffusion problems. The regularity of weak solution is firstly established by using variational approach and Mittag-Leffler function. Then several optimal error estimates are derived with low regularity data. Finally, numerical experiments are conducted to verify the theoretical results.
In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order $α\in(0,1)$ in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time le…
This article devotes to developing robust but simple correction techniques and efficient algorithms for a class of second-order time stepping methods, namely the shifted fractional trapezoidal rule (SFTR), for subdiffusion problems to resolve the initial singularity and nonlocality. The stability analysis and sharp error estimates in terms of the smoothness of the initial data and source term are presented. As a byproduct in numerical tests, we find amazingly that the Crank-Nicolson scheme ($θ=\frac{1}{2}$) without initial corrections can restore the optimal convergence rate for the subdiffusi…
The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order $α\in(0,1)$. After using quasi-boundary value method to regularize the "mildly" ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ) in time. We provide a thorough error analysis of the resulting discrete system in both cases of smooth and nonsmooth data. The analysis relies heavily on smoothing properties of (discrete) solut…
We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with th…
We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of a nonsmooth initial data. The diffusion coefficient is allowed to depend on time. It is well-known that in such parabolic problems the spatial error increases when time decreases. The rate of this time-dependency is a function of the fractional parameter and the regularity of the initial condition. We use the energy method to find optimal bounds on the error under weak and natural assumptions on the diffusivity. First, we prove the result for the linear problem and th…
A New $L2-1_σ$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface
School of Arts and Sciences, Guangzhou Maritime University（广州航海学院人文与科学学部）；School of Information Engineering, Zhejiang Ocean University（浙江海洋大学信息工程学院）；Fuzhou Vocational Technical College（抚州职业技术学院）
This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems. This method uses piecewise constant functions in the temporal discretization and continuous piecewise linear functions in the spatial discretization. Nearly optimal convergence rate with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate $ \ln(1/τ)(\sqrt{\ln(1/h)} h^2+τ)$ is derived under appropriate regularity assumption on the source term. Convergence is also established without smoothness assumption on the init…
In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional high-order numerical methods is inefficient. We try to overcome this difficulty in a novel approach by combining variable transformation techniques with spectral methods. The idea is to first use suitable variable transformation to re-scale the underlying equation, then construct spectral methods for the re-scaled equation. We establish a new variational fr…
Inverse problems of recovering space-dependent parameters, e.g., initial condition, space-dependent source or potential coefficient, in a subdiffusion model from the terminal observation have been extensively studied in recent years. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this work, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source and inverse potential problems, from the terminal observation at an unknown time. The subdiffusive nature of the…
In this article we study inverse problems of recovering a space-time dependent source component from the lateral boundary observation in a subidffusion model. The mathematical model involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time, and a second-order elliptic operator with time-dependent coefficients. We establish a well-posedness and a conditional stability result for the inverse problems using a novel perturbation argument and refined regularity estimates of the associated direct problem. Further, we present an algorithm for efficiently and accurately reconstru…
In this work, we study an inverse problem of recovering a space-time dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian-Caputo fractional derivative of order $α\in(0,1)$ in time. The main technical challenges of both theoretical and numerical analysis lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results usi…
Fractional Klein-Kramers equation can well describe subdiffusion in phase space. In this paper, we develop the fully discrete scheme for fractional Klein-Kramers equation based on the backward Euler convolution quadrature and local discontinuous Galerkin methods. Thanks to the obtained sharp regularity estimates in temporal and spatial directions after overcoming the hypocoercivity of the operator, the complete error analyses of the fully discrete scheme are built. % , the main challenge of which comes from the hypocoercivity of the operator. It's worth mentioning that the convergence of the p…
In this work, we consider the numerical recovery of a spatially dependent diffusion coefficient in a subdiffusion model from distributed observations. The subdiffusion model involves a Caputo fractional derivative of order $α\in(0,1)$ in time. The numerical estimation is based on the regularized output least-squares formulation, with an $H^1(Ω)$ penalty. We prove the well-posedness of the continuous formulation, e.g., existence and stability. Next, we develop a fully discrete scheme based on the Galerkin finite element method in space and backward Euler convolution quadrature in time. We prove…
Numerical Analysis of Space-Time Dependent Source Identification in Subdiffusion Equations
Univ Rouen Normandie, CNRS, Normandie Univ, LMRS UMR 6085（鲁昂诺曼底大学）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
In this paper, we introduce a fourth-order accurate finite element method for incompressible variable density flow. The method is implicit in time and constructed with the Taylor series technique, and uses standard high-order Lagrange basis functions in space. Taylor series time-stepping relies on time derivative correction terms to achieve high-order accuracy. We provide detailed algorithms to approximate the time derivatives of the variable density Navier-Stokes equations. Numerical validations confirm a fourth-order accuracy for smooth problems. We also numerically illustrate that the Taylo…
In this work, we present a second-order nonuniform time-stepping scheme for the time-fractional Allen-Cahn equation. We show that the proposed scheme preserves the discrete maximum principle, and by using the convolution structure of consistency error, we present sharp maximum-norm error estimates which reflect the temporal regularity. As our analysis is built on nonuniform time steps, we may resolve the intrinsic initial singularity by using the graded meshes. Moreover, we propose an adaptive time-stepping strategy for large time simulations. Numerical experiments are presented to show the ef…
Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo's derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time-steps are always incorporated into our approaches so that an adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity…
The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška--Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it d…
