Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

Discretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions

In this work, we propose novel discretisations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretisations with order of convergence depending on the regularity of the domain and the function on which the fractional Laplacian is acting. Unlike other existing approaches in literature, our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.FdT acknowledges support of Toppforsk project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway. ERCIM ``Alain Benoussan" Fellowship programm

Space-time Methods for Time-dependent Partial Differential Equations

Modern discretizations of time-dependent PDEs consider the full problem in the space-time cylinder and aim to overcome limitations of classical approaches such as the method of lines (first discretize in space and then solve the resulting ODE) and the Rothe method (first discretize in time and then solve the PDE). A main advantage of a holistic space-time method is the direct access to space-time adaptivity and to the backward problem (required for the dual problem in optimization or error control). Moreover, this allows for parallel solution strategies simultaneously in time and space. Several space-time concepts where proposed (different conforming and nonconforming space-time finite elements, the parareal method, wavefront relaxation etc.) but this topic has become a rapidly growing field in numerical analysis and scientific computing. In this workshop the focus is the development of adaptive and flexible space-time discretization methods for solving parabolic and hyperbolic space-time partial differential equations

Numerical Analysis of Fractional-Order Differential Equations with Nonsmooth Data

This thesis is devoted to theoretical and experimental justifications of numerical methods for fractional differential equations, which have received significant attention over the past decades due to their extraordinary capability of modeling the dynamics of anomalous diffusion processes. In recent years, a number of numerical schemes were developed, analyzed and tested. However, in most of these interesting works, the error estimates were established under the assumption that the solution is sufficiently smooth. Unfortunately, it has been shown that these assumptions are too restrictive for the solutions of fractional differential equations. The goal of this thesis is to develop robust numerical schemes and to establish optimal error bounds that are expressed directly in terms of the regularity of the problem data. We are especially interested in the case of nonsmooth data arising in many applications, e.g. inverse and control problems. After some background introduction and preliminaries in Chapters 1 and 2, we analyze two semidiscrete schemes obtained by standard Galerkin finite element approximation and lumped mass finite element method in Chapter 3. The error bounds for approximate solutions of the homogeneous and inhomogeneous problems are established separately in terms of the smoothness of the data directly. In Chapter 4 we revisit the most popular fully discrete scheme based on L1-type approximation in time and Galerkin finite element method in space and show the first order convergence in time by the discrete Laplace transform technique, which fills the gap between the existing error analysis theory and numerical results. Two fully discrete schemes based on convolution quadrature are developed in Chapter 5. The error bounds are given using two different techniques, i.e., discretized operational calculus and discrete Laplace transform. Last, in Chapter 6, we summarize our work and mention possible future research topics. In each chapter, the discussion is focused on the fractional diffusion model and then extended to some other fractional models. Throughout, numerical results for one- and two-dimensional examples will be provided to illustrate the convergence theory

Numerical Approximation of Partial Differential Equations Involving Fractional Differential Operators

The negative powers of an elliptic operator can be approximated via its Dunford-Taylor integral representation, i.e. we approximate the Dunford-Taylor integral with an exponential convergent sinc quadrature scheme and discretize the integrand (a diffusion-reaction problem) at each quadrature point using the finite element method. In this work, we apply this discretization strategy for a parabolic problem involving fractional powers of elliptic operators and a stationary problem involving the integral fractional Laplacian. The approximation of the parabolic problem is twofold: the homogenous problem and the non-homogeneous problem. We propose an approximation scheme for the homogeneous problem based on a complex-valued integral representation of the solution operator. An exponential convergent sinc quadrature scheme with a hyperbolic contour and a complex-valued finite element method are developed. The approximation of the non-homogeneous problem in space follows the same idea from the homogeneous problem but we need to additionally discretize the problem in the time domain. Here we consider two different approaches: a pseudo-midpoint quadrature scheme in time based on Duhamel’s principle and the Crank-Nicolson time stepping method. Both methods guarantee second order convergence in time but require different sinc quadrature schemes to approximate the corresponding fractional operators. The time stepping method is stable provided that the sinc quadrature spacing is sufficiently small. In terms of the approximation of the stationary problem involving integral fractional Laplacian, we consider a Dunford-Taylor integral representation of the bilinear form in the weak formulation. After approximating the integral with a sinc quadrature scheme, we need to approximate the integrand at each quadrature point which contains a solution of a diffusion-reaction equation defined on the whole space. We approximate the integrand problem on a truncated domain together with the finite element method. For both problems, we provide L² error estimates between solutions and their final approximations. Numerical implementation and results illustrating the behavior of the algorithms are also provided

Coefficient Identification Problems in Heat Transfer

The aim of this thesis is to find the numerical solution for various coefficient identification problems in heat transfer and extend the possibility of simultaneous determination of several physical properties. In particular, the problems of coefficient identification in a fixed or moving domain for one and multiple unknowns are investigated. These inverse problems are solved subject to various types of overdetermination conditions such as non-local, heat flux, Cauchy data, mass/energy specification, general integral type overdetermination, time-average condition, time-average of heat flux, Stefan condition and heat momentum of the first and second order. The difficulty associated with these problems is that they are ill-posed, as their solutions are unstable to inclusion of random noise in input data, therefore traditional techniques fail to provide accurate and stable solutions. Throughout this thesis, the Crank-Nicolson finite-difference method (FDM) is mainly used as a direct solver except in Chapter 7 where a three-level scheme is employed in order to deal with the nonlinear heat equation. An explicit FDM scheme is also employed in Chapter 10 for the two-dimensional case. The inverse problems investigated are discretised using the FDM and recast as nonlinear least-squares minimization problems with simple bounds on the unknown coefficients. The resulting problem is efficiently solved using the \emph{fmincon} or \emph{lsqnonlin} routines from MATLAB optimization toolbox. The Tikhonov regularization method is included where necessary. The choice of the regularization parameter(s) is thoroughly discussed. The stability of the numerical solution is investigated by introducing Gaussian random noise into the input data. The numerical solutions are compared with their known analytical solution, where available, and with the corresponding direct problem numerical solution where no analytical solution is available