Background

Today, we will solve the time-dependent convection-diffusion equation,

$$\begin{array}{rcl} \dot{u} + b \cdot \nabla u - \nabla \cd... ..., \\ u(\cdot,0) &=& u_0 \quad \mbox{in } \Omega, \end{array}$$ (1)

where the convection is given by the vector $b = b(x,t)$ and the diffusion is given by $a = a(x,t)$. The $\mathrm{dG}(0)$ formulation of the convection-diffusion equation is given by

$$\begin{array}{ll} \int_{\Omega} U_n v \, dx + k \int_{\Om... ...{\Omega} U_{n-1} v \, dx \quad \forall v \in V_h, \end{array}$$ (2)

where $k = t_n - t_{n-1}$ denotes the size of the time step, $U_n$ denotes the value at $t=t_n$, and $U_{n-1}$ denotes the value at $t=t_{n-1}$. Note that also $\gamma$, $g_D$, and $g_N$ should be evaluated at $t=t_n$. The $\mathrm{dG}(0)$ method is also known as the backward Euler method.

Before today's computer session, make sure that you understand and can answer the following questions.

Question 1 Derive the variational formulation (2) from the convection-diffusion equation (1).

Question 2 How does the corresponding variational formulation look for the $\mathrm{cG}(1)$ method? This method is also known as the Crank-Nicolson method.

Question 3 Verify that

$$b(x,t) = (-(x_2-c_2), x_1-c_1),$$ (3)

where $c = (c_1,c_2)$ is a constant, is divergence-free, i.e., $\nabla \cdot b = 0$. Draw a simple sketch of the vector-field given by $b$. (Draw some arrows on a piece of paper.) Why is it important that $b$ is divergence-free?