A Posteriori Error Estimates for Fully Discrete Finite Element Method for Generalized Diffusion Equation with Delay

Abstract

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.

Recovering critical parameter for nonlinear Allen–Cahn equation by fully discrete continuous data assimilation algorithms *

“…Although many researchers have investigated the stability and convergence of some numerical methods for nonlinear Allen-Cahn equation because of the importance of the equation in the modeling of phase transitions and interfacial dynamics in materials science, the topic of the parameter recovery for such type of equation remains unexplored. In our previous paper, the uniform-in-time error estimates for the implicit Euler and BDF2 finite element approximation of reaction-diffusion equation with unknown initial condition, but with a known critical parameter ϵ 0 , has been derived in [64]. In this paper, we presented and analyzed a new way to recover the unknown critical parameter of a dissipative system using fully discrete continuous DA algorithm for the Allen-Cahn equation.…”

“…[13,36,53]). We also obtained the uniform-in-time error estimates for the implicit backward differential formula (BDF) finite element approximation of Allen-Cahn equation with unknown initial condition, but with a known critical parameter ϵ 0 [64]. Although numerical and theoretical analysis of DA algorithms for dissipative PDEs is an active research area, few studies have been done on the parameter recovery by continuous DA algorithms [17].…”

Reconstruction-Based a Posteriori Error Estimates for the L1 Method for Time Fractional Parabolic Problems

“…Such computable a posteriori error estimates have been investigated by many researchers for various numerical methods for integer-order parabolic problems during the last decades (see, e.g. [1,5,37,47,[50][51][52][53]). A posteriori error estimates and adaptivity are now in many cases very successful tools for efficient numerical computations of linear as well as nonlinear integer-order problems.…”

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

“…Indeed, there still exists no work focusing on the superconvergence analysis of the delay model, let alone the nonlinear system (1). For the FEM of the delay model, interested authors can refer to [45,46].…”