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We propose and analyze a mixed finite element method for the spatial approximation of a time-fractional Fokker–Planck equation in a convex polyhedral domain, where the given driving force is a function of space. Taking into account the limited smoothing properties of the model, and considering an appropriate splitting of the errors, we employed a sequence of clever energy arguments to show optimal convergence rates with respect to both approximation properties and regularity results. In particular, error bounds for both primary and secondary variables are derived in L 2 -norm for cases with smooth and nonsmooth initial data. We further investigate a fully implicit time-stepping scheme based on a convolution quadrature in time generated by the backward Euler method. Our main result provides pointwise-in-time optimal L 2 -error estimates for the primary variable. Numerical examples are then presented to illustrate the theoretical contributions. [ABSTRACT FROM AUTHOR]

We consider a time-fractional biharmonic equation involving a Caputo derivative in time of fractional order α ∈ (0 , 1) and a locally Lipschitz continuous nonlinearity. Local and global existence of solutions is discussed and detailed regularity results are provided. A finite element method in space combined with a backward Euler convolution quadrature in time is analyzed. Our objective is to allow initial data of low regularity compared to the number of derivatives occurring in the governing equation. Using a semigroup type approach, error estimates of optimal order are derived for solutions with smooth and nonsmooth initial data. Numerical tests are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a mixed finite element method is applied in spatial directions while keeping time variable continuous to a class of time-fractional diffusion problems with time-dependent coefficients on a bounded convex polygonal domain. Based on an energy argument combined with a repeated application of an integral operator, optimal error estimates, which are optimal with respect to both approximation properties and regularity results, are derived for the semidiscrete problem with smooth as well as nonsmooth initial data. Specially, a priori error bounds for both primary and secondary variables in L 2 -norm are established. Since the comparison between Fortin projection and the mixed Galerkin approximation of the secondary variable yields an improved rate of convergence, therefore, as a by-product, we derive L p -estimates for the error in primary variable. Finally, some numerical experiments are conducted to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We derive optimal L 2 -error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order α ∈ (0 , 1) , for cases with smooth and nonsmooth initial data. A general framework is introduced allowing a unified error analysis of Galerkin type space approximation methods. The analysis is based on a semigroup type approach and exploits the properties of the inverse of the associated elliptic operator. Completely discrete schemes are analyzed in the same framework using a backward Euler convolution quadrature method in time. Numerical examples including conforming, nonconforming and mixed finite element methods are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider the numerical approximation of a generalized fractional Oldroyd-B fluid problem involving two Riemann-Liouville fractional derivatives in time. We establish regularity results for the exact solution which play an important role in the error analysis. A semidiscrete scheme based on the piecewise linear Galerkin finite element method in space is analyzed, and optimal with respect to the data regularity error estimates are established. Further, two fully discrete schemes based on convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are investigated and related error estimates for smooth and nonsmooth data are derived. Numerical experiments are performed with different values of the problem parameters to illustrate the efficiency of the method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, the piecewise-linear finite element method (FEM) is applied to approximate the solution of time-fractional diffusion equations on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal error bounds in $$L^2(\varOmega )$$ -, $$H^1(\varOmega )$$ -norms, and a quasi-optimal bound in $$L^{\infty }(\varOmega )$$ -norm are derived for the semidiscrete FEM for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a $$t^m$$ type of weights to take care of the singular behavior of the continuous solution at $$t=0$$ . The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L-norm error over a finite time interval converges optimally as O( h+Δ t), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and Δ t the time step. [ABSTRACT FROM AUTHOR]

A general procedure to construct ADI methods for multidimensional problems was originated by Beam and Warming using the method of approximate factorization. In this paper, we extend the method of approximate factorization to solve a viscous wave equation. The method can be combined with any implicit linear multistep method for the time integration of the wave equation. The stability of the factored schemes and their underlying schemes is analyzed based on a discrete Fourier analysis and the energy method. Convergence proofs are presented and numerical results supporting our analysis are provided. [ABSTRACT FROM AUTHOR]

We derive a fourth-order compact finite difference scheme for a two-dimensional elliptic problem with a mixed derivative and constant coefficients. We conduct experimental study on numerical solution of the problem discretized by the present compact scheme and the traditional second-order central difference scheme. We study the computed accuracy achieved by each scheme and the performance of the Gauss-Seidel iterative method, the preconditioned GMRES iterative method, and the multigrid method, for solving linear systems arising from the difference schemes. [ABSTRACT FROM AUTHOR]