Problem $3(35$ pts$).$ A rectangular channel of size $L_1{L}_2$ connects two tanks. Tanks $1$

and $2$ contain sugar in water solution, with concentrations $c_1$ and $c_2,$ respectively. There is

no flux of mass through the bottom and top sides of the channel $(J_n=0).$

$($a$)[5$pts$]$ The steady$-$state sugar concentration $c(x,y)$ in the channel satisfies the following

governing equations:

div$(J)=0(mass$ balance$)$

$J=-$Dgrad$(c)(Fick\text{'}s$ law $of$ diffusion$)$

Here $J$ is the vector of mass flux of sugar in the channel, $D$ is the diffusion coefficient

of sugar in water, while div and grad stand for the divergence and gradient differential

operators in two dimensions, respectively.

Combine equations $(6)$ and $(7)$ to obtain a second$-$order partial differential equation

$($PDE$)$ for the concentration $c(x,y).$

$($b$)10pts$ List appropriate boundary conditions on each side of the channel. Together with

these boundary conditions, the PDE found in $($a$)$ defines a boundary value problem

$($BVP$).$ Use the method of separation of variables to solve this BVP$.$ Hint: you can

use an additive separation ansatz in the form $c(x,y)=c_x(x)+c_y(y).$

$($c$)[10$pts$]$ The BVP defined in $($b$)$ is formulated in strong form. A necessary step to

implement the BVP within the Finite Element Method $($FEM$)$ is to derive the weak

form of the BVP$.$ In the Galerkin method of weighted residuals, this is achieved by

multiplying the governing PDE with an arbitrary test function tilde$(c),$ by integrating over

the domain, and by lowering the order of differentiation via integration by parts. Derive

a suitable weak form of the BVP defined in $($b$).$

$($d$)[5$pts$]$ Briefly explain how the FEM implementation of the weak form found in $($c$)$ leads

to a discrete system of equations. Include the following concepts in your explanation:

mesh, elements, nodes, shape functions, quadrature, assembly, "stiffness" matrix.

$($e$)[5$pts$]$ The global system of equations obtained in $($d$)$ is sparse and symmetric positive$-$

definite. Suggest suitable direct and iterative solution methods for such a system of

equations.