Problem $3.$ The two$-$dimensional square plate in the figure below has edges of length $L.$ The

steady$-$state temperature distribution $T(x,y)$ in the plate satisfies the following governing

equations:

div$(q)=0(energy$ balance$)$

$q=-$kgrad$(T)(Fourier$ law $of$ heat conduction$)$

Here $q$ is the vector of heat flux, $k$ is the constant of thermal conductivity, while div and grad

stand for the divergence and gradient differential operators in two dimensions, respectively.

$($a$)$ Combine equations. $(3)$ and $(4)$ to obtain a second$-$order partial differential equation

$($PDE$)$ for the temperature $T(x,y).$

$($b$)$ The temperature on the left side of the plate is kept at $T_0,$ while the temperature

on the right side of the plate is kept at $T_1.$ There is no normal heat flux on the

bottom and top sides $(q_n=0).$ 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 $T(x,y)=T_x(x)+T_y(y).$

$($c$)$ 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$(T),$ 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$)$ Explain how the FEM implementation of the weak form found in $($c$)$ leads to a dis$-$

crete system of equations. Include the following concepts in your explanation: mesh,

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

$($e$)$ The global system of equations obtained in $($d$)$ is sparse and symmetric positive$-$definite.

Suggest a suitable iterative solution method for such system and sketch its main algo$-$

rithm.