Solution of the Cauchy problem for the Brinkman equations using an alternating method of fundamental solutions

Abstract

In this paper, we intend to formulate and solve Cauchy problems for the Brinkman equations governing the flow of fluids in porous media, which have never been investigated before in such an inverse formulation. The physical scenario corresponds to situations where part of the boundary of the fluid domain is hostile or inaccessible, whilst on the remaining friendly part of the boundary we prescribe or measure both the fluid velocity and traction. The resulting mathematical formulation leads to a linear but ill-posed problem. A convergent algorithm based on solving two sub-sequences of mixed direct problems is developed. The direct solver is based on the method of fundamental solutions which is a meshless boundary collocation method. Since the investigated problem is ill-posed, the iterative process is stopped according to the discrepancy principle at a threshold given by the amount of noise with which the input measured data is contaminated in order to prevent the manifestation of instability. Results inverting both exact and noisy data for two- and three-dimensional problems demonstrate the convergence and stability of the proposed numerical algorithm.