The backward Euler anisotropic a posteriori error analysis for parabolic integro-differential equations (Q2821183)

This paper report about the a posteriori error analysis of the approximate solutions of the initial value and boundary value problem for parabolic integro-differential equation NEWLINENEWLINE\[NEWLINE u_{t}(x,t)+ Au(x,t)= \int_{o}^{t}{B}(t,s)u(x,s))ds + f(x,t), \qquad (x,t)\in\Omega \times(0,T),NEWLINE\]NEWLINE NEWLINEwhere \(\Omega\in \mathbb{R}^{2}\), \({A}u =-\nabla(A\nabla u)\), \({B}u =-\nabla(B(t,s)\nabla u)\). The discretization by space variables use the finite elements method and implicit Euler method for the variable \(t\). For the Volterra integral term it is used the linear approximation. Analysis of the error consist to construct two estimators. The numerical results are given considering the particulary case \(A = B = I\).