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

Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates

We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L2-norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetic

Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity. Read More: http://epubs.siam.org/doi/10.1137/14097956