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

Lumped finite elements for reaction–cross-diffusion systems on stationary surfaces

We consider a lumped surface finite element method (LSFEM) for the spatial approximation of reaction–diffusion equations on closed compact surfaces in R3R3 in the presence of cross-diffusion. We provide a fully-discrete scheme by applying the Implicit–Explicit (IMEX) Euler method. We provide sufficient conditions for the existence of polytopal invariant regions for the numerical solution after spatial and full discretisations. Furthermore, we prove optimal error bounds for the semi- and fully-discrete methods, that is the convergence rates are quadratic in the meshsize and linear in the timestep. To support our theoretical findings, we provide two numerical tests. The first test confirms that in the absence of lumping numerical solutions violate the invariant region leading to blow-up due to the nature of the kinetics. The second experiment is an example of Turing pattern formation in the presence of cross-diffusion on the sphere

A moving grid finite element method applied to a mechanobiochemical model for 3D cell migration

This work presents the development, analysis and numerical simulations of a biophysical model for 3D cell deformation and movement, which couples biochemical reactions and biomechanical forces. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled by a force balancing equation for the displacements, the pressure and contractile forces are driven by actin and myosin dynamics, and these are in turn modelled by a system of reaction-diffusion equations on a moving cell domain. The biophysical model consists of highly non-linear partial differential equations whose analytical solutions are intractable. To obtain approximate solutions to the model system, we employ the moving grid finite element method. The numerical results are supported by linear stability theoretical results close to bifurcation points during the early stages of cell migration. Numerical simulations exhibited show both simple and complex cell deformations in 3-dimensions that include cell expansion, cell protrusion and cell contraction. The computational framework presented here sets a strong foundation that allows to study more complex and experimentally driven reaction-kinetics involving actin, myosin and other molecular species that play an important role in cell movement and deformation

On Preservation of Positivity in Some Finite Element Methods for the Heat Equation

We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We complement in a number of ways earlier studies of the possible extension of this fact to spatially semidiscrete and fully discrete piecewise linear finite element discretizations, based on the standard Galerkin method, the lumped mass method, and the finite volume element method. We also provide numerical examples that illustrate our findings