Volume 17, Issue 4 - Contains the First Part of the Special CMAM Issue on Numerical Analysis of Fractional Differential Equations, Guest Editors: Raytcho Lazarov, Petr Vabishchevich, and Bangti Jin

The dynamics of elastic media, constrained by Dirichlet boundary conditions, can be modeled as an operator DAE of semi-explicit structure. These models include flexible multibody systems as well as applications with boundary control. In order to use adaptive methods in space, we analyze the properties of the Rothe method concerning stability and convergence for this kind of systems. We consider a regularization of the operator DAE and prove the weak convergence of the implicit Euler scheme. Furthermore, we consider perturbations in the semi-discrete systems which correspond to additional errors such as spatial discretization errors.

We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

In this work, we design a posteriori error estimation and mesh adaptivity for multiple goal functionals. Our method is based on a dual-weighted residual approach in which localization is achieved in a variational form using a partition-of-unity. The key advantage is that the method is simple to implement and backward integration by parts is not required. For treating multiple goal functionals we employ the adjoint to the adjoint problem (i.e., a discrete error problem) and suggest an alternative way for its computation. Our algorithmic developments are substantiated for elliptic problems in terms of four different numerical tests that cover various types of challenges, such as singularities, different boundary conditions, and diverse goal functionals. Moreover, several computations with higher-order finite elements are performed.

This paper studies the mixed element method for the boundary value problem of the biharmonic equation Δ2⁢u=f{\Delta^{2}u=f} in two dimensions. We start from a u∼∇⁡u∼∇2⁡u∼div⁢∇2⁡u{u\sim\nabla u\sim\nabla^{2}u\sim\operatorname{div}\nabla^{2}u} formulation that is discussed in [4] and construct its stability on H01⁢(Ω)×H~01⁢(Ω)×L¯sym2⁢(Ω)×H-1⁢(div,Ω){H^{1}_{0}(\Omega)\times\tilde{H}^{1}_{0}(\Omega)\times\bar{L}_{\mathrm{sym}}^% {2}(\Omega)\times H^{-1}(\operatorname{div},\Omega)}. Then we utilise the Helmholtz decomposition of H-1⁢(div,Ω){H^{-1}(\operatorname{div},\Omega)} and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given.

In this study, we develop and implement numerical schemes to solve classes of two-dimensional fourth-order partial differential equations. These methods are fourth-order accurate in space and second-order accurate in time and require only nine spatial grid points of a single compact cell. The proposed discretizations allow the use of Dirichlet boundary conditions only without the need to discretize the derivative boundary conditions and thus avoids the use of ghost points. No transformation or linearization technique is used to handle nonlinearity and the obtained block tri-diagonal nonlinear system has been solved by Newton’s block iteration method. It is discussed how our formulation is able to tackle linear singular problems and it is ensured that the methods retain their orders and accuracy everywhere in the solution region. The proposed two-level method is shown to be unconditionally stable for a class of two-dimensional fourth-order linear parabolic equation. We also discuss the alternating direction implicit (ADI) method for solving two-dimensional fourth-order linear parabolic equation. The proposed difference methods has been successfully tested on the two-dimensional vibration problem, Boussinesq equation, extended Fisher–Kolmogorov equation and Kuramoto–Sivashinsky equation. Numerical results demonstrate that the schemes are highly accurate in solving a large class of physical problems.

In this paper, an a priori estimate for the corresponding differential problem is obtained by using the method of the energy inequalities. We construct a difference analog of the Caputo fractional derivative with generalized memory kernel (L1 formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the (2-α){(2-\alpha)}-th order in time for the generalized time-fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid L2{L_{2}}-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by numerical calculations carried out for some test problems.

Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. The numerical solution of corresponding model problems via a spectral method is analyzed. Its efficiency and features of the model problems are illustrated by numerical experiments.

In this paper, we develop a numerical scheme for the space-time fractional parabolic equation, i.e. an equation involving a fractional time derivative and a fractional spatial operator. Both the initial value problem and the non-homogeneous forcing problem (with zero initial data) are considered. The solution operator E⁢(t){E(t)} for the initial value problem can be written as a Dunford–Taylor integral involving the Mittag-Leffler function eα,1{e_{\alpha,1}} and the resolvent of the underlying (non-fractional) spatial operator over an appropriate integration path in the complex plane. Here α denotes the order of the fractional time derivative. The solution for the non-homogeneous problem can be written as a convolution involving an operator W⁢(t){W(t)} and the forcing function F⁢(t){F(t)}. We develop and analyze semi-discrete methods based on finite element approximation to the underlying (non-fractional) spatial operator in terms of analogous Dunford–Taylor integrals applied to the discrete operator. The space error is of optimal order up to a logarithm of 1h{\frac{1}{h}}. The fully discrete method for the initial value problem is developed from the semi-discrete approximation by applying a sinc quadrature technique to approximate the Dunford–Taylor integral of the discrete operator and is free of any time stepping. The sinc quadrature of step size k involves k-2{k^{-2}} nodes and results in an additional O⁢(exp⁡(-ck)){O(\exp(-\frac{c}{k}))} error. To approximate the convolution appearing in the semi-discrete approximation to the non-homogeneous problem, we apply a pseudo-midpoint quadrature. This involves the average of Wh⁢(s){W_{h}(s)}, (the semi-discrete approximation to W⁢(s){W(s)}) over the quadrature interval. This average can also be written as a Dunford–Taylor integral. We first analyze the error between this quadrature and the semi-discrete approximation. To develop a fully discrete method, we then introduce sinc quadrature approximations to the Dunford–Taylor integrals for computing the averages. We show that for a refined grid in time with a mesh of O⁢(𝒩⁢log⁡(𝒩)){O({\mathcal{N}}\log({\mathcal{N}}))} intervals, the error between the semi-discrete and fully discrete approximation is O⁢(𝒩-2+log⁡(𝒩)⁢exp⁡(-ck)){O({\mathcal{N}}^{-2}+\log({\mathcal{N}})\exp(-\frac{c}{k}))}. We also report the results of numerical experiments that are in agreement with the theoretical error estimates.

A nonlocal convection-diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model parameters, the nonlocal steady and unsteady state master equations are shown to be well-posed in a weak sense. Then the nonlocal operator is shown to be the generator of finite-range nonsymmetric jump processes and, when certain conditions on the model parameters hold, the generators of finite and infinite activity Lévy and Lévy-type jump processes are shown to be special instances of the nonlocal operator.