RESUMEN
In this paper, we derive several a posteriori error estimators for generalized diffusion equation with delay in a convex polygonal domain. The Crank-Nicolson method for time discretization is used and a continuous, piecewise linear finite element space is employed for the space discretization. The a posteriori error estimators corresponding to space discretization are derived by using the interpolation estimates. Two different continuous, piecewise quadratic reconstructions are used to obtain the error due to the time discretization. To estimate the error in the approximation of the delay term, linear approximations of the delay term are used in a crucial way. As a consequence, a posteriori upper and lower error bounds for fully discrete approximation are derived for the first time. In particular, long-time a posteriori error estimates are obtained for stable systems. Numerical experiments are presented which confirm our theoretical results.
RESUMEN
In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain L2(H1)-L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain L2(L2)-L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control.
RESUMEN
Understanding how patterns and travelling waves form in chemical and biological reaction-diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction-diffusion models, such as Lotka-Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction-diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.