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.

Appendix

In three dimensions, we approximate the fluid velocity ${\varvec{u} }= ( u_1, u_2, u_3)$ and the pressure p by ${\varvec{u} }_N= ( u_{N_1}, u_{N_2}, u_{N_3})$ and $p_N$, respectively, where

$$\begin{aligned} u_{N_i}({\varvec{x} })=&\sum _{j=1}^{N} \left( \alpha _j G_{i1}({\varvec{x} },{\varvec{x} }^\prime _j)+ \beta _j G_{i2}({\varvec{x} },{\varvec{x} }^\prime _j) + \gamma _j G_{i3}({\varvec{x} },{\varvec{x} }^\prime _j)\right) ,\nonumber \\ i&=1,2,3, \quad {\varvec{x} } \in \overline{\Omega }, \end{aligned}$$

(A.1)

and the pressure p by $p_N$ where

$$\begin{aligned} p_{N} ({\varvec{x} })=\sum _{j=1}^{N} \left( \alpha _j P_{1}({\varvec{x} },{\varvec{x} }^\prime _j)+ \beta _j P_{2}({\varvec{x} },{\varvec{x} }^\prime _j) + \gamma _j P_{3}({\varvec{x} },{\varvec{x} }^\prime _j) \right) , \quad {\varvec{x} } \in \overline{\Omega }, \end{aligned}$$

(A.2)

Fig. 13

Typical node distributions on sphere

Fig. 14

Example 3: Results for ${\varvec{u} }$, ${\varvec{t} }$ and p along the circle $x^2+y^2=3/4$ on the surface $\Gamma _1$ obtained after 500 iterations in case of no noise

where

$$\begin{aligned} G_{ik}({\varvec{x} }, {\varvec{x} }^\prime )=\dfrac{1}{4 \pi \mu \kappa ^2 r^3} \left[ \left( -1+\left( 1+\kappa r +\kappa ^2 r^2 \right) \textrm{e}^{-\kappa r}\right) \delta _{ik} \right. \nonumber \\ \left. + \dfrac{(x_i-x_i^\prime ) (x_k-x_k^\prime ) }{r^2} \left( 3 - \left( 3+3 \kappa r +\kappa ^2 r^2\right) \textrm{e}^{-\kappa r} \right) \right] , \quad i,k=1,2,3, \end{aligned}$$

(A.3)

$$\begin{aligned} P_k({\varvec{x} }, {\varvec{x} }^\prime )= \dfrac{x_k-x_k^\prime }{4 \pi r^3}, \quad k=1,2,3, \end{aligned}$$

(A.4)

is the fundamental solution of the three-dimensional Brinkman and continuity equations (2.2a), see, e.g., [23, 24]. The approximation for the fluid traction ${\varvec{t} }=(t_1,t_2, t_3)$ is ${\varvec{t} }_N=(t_{N_1},t_{N_2}, t_{N_3})$, where

$$\begin{aligned} t_{N_i}({\varvec{x} })=\sum _{j=1}^{N} \left( \alpha _j D_{i1}({\varvec{x} },{\varvec{x} }^\prime _j)+ \beta _j D_{i2}({\varvec{x} },{\varvec{x} }^\prime _j) + \gamma _j D_{i3}({\varvec{x} },{\varvec{x} }^\prime _j) \right) , \quad i=1,2,3, \quad {\varvec{x} } \in \partial \Omega , \end{aligned}$$

(A.5)

with

$$\begin{aligned} D_{i\ell }=-P_{\ell }\, n_i+\mu \sum _{k=1}^3 \left( \dfrac{\partial G_{i\ell }}{\partial x_k} + \dfrac{\partial G_{k\ell }}{\partial x_i} \right) n_k, \quad i, \ell = 1,2,3. \end{aligned}$$

(A.6)

Fig. 15

Example 3: Results for ${\varvec{u} }$, ${\varvec{t} }$ and p along the circle $x^2+y^2=3/4$ on the surface $\Gamma _1$, noise $\textrm{p}=1 \%$, obtained after 63 iterations given by the stopping criterion (7.12)

Fig. 16

Example 3: Results for ${\varvec{u} }$, ${\varvec{t} }$ and p along the circle $x^2+y^2=3/4$ on the surface $\Gamma _1$, noise $\textrm{p}=3 \%$, obtained after 33 iterations given by the stopping criterion (7.12)

Fig. 17

Example 3: Results for ${\varvec{u} }$, ${\varvec{t} }$ and p along the circle $x^2+y^2=3/4$ on the surface $\Gamma _1$, noise $\textrm{p}=5 \%$, obtained after 28 iterations given by the stopping criterion (7.12)

The partial derivatives needed in (A.6) are given by [24],

$$\begin{aligned} \dfrac{\partial G_{ik}}{\partial x_j} = \dfrac{1}{4 \pi \mu \kappa ^2 r^5} \left\{ (x_j-x_j^\prime ) \left[ 3-\left( 3 +3\kappa r +2\kappa ^2r^2 +\kappa ^{3} r^{3} \right) \textrm{e}^{-\kappa r}\right] \delta _{ik} \right. \end{aligned}$$

$$\begin{aligned} +\left( (x_k-x_k^\prime )\delta _{ij}+(x_i-x_i^\prime )\delta _{jk}\right) \left[ 3-\left( 3 +3\kappa r +\kappa ^2r^2\right) \textrm{e}^{-\kappa r} \right] \end{aligned}$$

$$\begin{aligned} \left. +\dfrac{(x_i-x_i^\prime )(x_j-x_j^\prime )(x_k-x_k^\prime )}{r^2}\left[ -15+\left( 15+15 \kappa r+6 \kappa ^2 r^2 +\kappa ^3 r^3\right) \textrm{e}^{-\kappa r} \right] \right\} , \quad i, j, k =1, 2, 3. \end{aligned}$$

(A.7)