Continuous data assimilation for displacement in a porous medium

Abstract

In this paper we propose the use of a continuous data assimilation algorithm for miscible flow models in a porous medium. In the absence of initial conditions for the model, observed sparse measurements are used to generate an approximation to the true solution. Under certain assumption of the sparse measurements and their incorporation into the algorithm it can be shown that the resulting approximate solution converges to the true solution at an exponential rate as time progresses. Various numerical examples are considered in order to validate the suitability of the algorithm.

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

“…Much attention has been devoted to the error estimates, with unknown initial condition, of semi-discrete finite element DA algorithms (see, e.g. [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].…”