Continuous data assimilation for the three-dimensional Brinkman–Forchheimer-extended Darcy model. (9th March 2016)

• Record Type: Journal Article
• Authors: Markowich, Peter A; Titi, Edriss S; Trabelsi, Saber
• Is Part Of: Nonlinearity. Volume 29:Number 4(2016:Apr.)
• Page Start: 1292
• Page End: 1328
• Publication Date: 2016-03-09
• Subjects: Brinkman–Forchheimer-extended Darcy model -- data assimilation -- down-scaling; 35Q35 -- 76B03 -- 86A10; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/0951-7715/29/4/1292
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper we introduce and analyze an algorithm for continuous data assimilation for a three-dimensional Brinkman–Forchheimer-extended Darcy (3D BFeD) model of porous media. This model is believed to be accurate when the flow velocity is too large for Darcy's law to be valid, and additionally the porosity is not too small. The algorithm is inspired by ideas developed for designing finite-parameters feedback control for dissipative systems. It aims to obtain improved estimates of the state of the physical system by incorporating deterministic or noisy measurements and observations. Specifically, the algorithm involves a feedback control that nudges the large scales of the approximate solution toward those of the reference solution associated with the spatial measurements. In the first part of the paper, we present a few results of existence and uniqueness of weak and strong solutions of the 3D BFeD system. The second part is devoted to the convergence analysis of the data assimilation algorithm.