= Solution
Use a <conforming finite element space> $V_h\subset H_0^1(\Omega)$ with the prescribed <basis>, and write $u_h(x,y,t)=\sum_jU_j(t)\phi_j(x,y)$. Testing the <advection-diffusion equation> by $\phi_i$ and applying <integration by parts> to its diffusion term gives
$$
\sum_j\left(\int_\Omega\phi_j\phi_i\right)\dot U_j
=\sum_j\left[\int_\Omega\phi_i(\partial_x\phi_j+\partial_y\phi_j)-\int_\Omega\nabla\phi_i\cdot\nabla\phi_j\right]U_j.
$$
Thus
$$
\boxed{M\dot U=PU,\quad M_{ij}=\int_\Omega\phi_i\phi_j,\quad
P_{ij}=\int_\Omega\phi_i(\partial_x\phi_j+\partial_y\phi_j)-\int_\Omega\nabla\phi_i\cdot\nabla\phi_j.}
$$
Here $M$ is the <mass matrix>; it is a <symmetric positive-definite matrix> because the <basis> is <linearly independent>. The initial vector represents the chosen finite element approximation to $g$, for example its <orthogonal projection>.
For any coefficient vector $U$, put $v_h=\sum_jU_j\phi_j$. The <skew symmetry of Galerkin advection> with constant drift $(1,1)$ gives
$$
\int_\Omega v_h(\partial_xv_h+\partial_yv_h)
=\frac12\int_{\partial\Omega}v_h^2(n_x+n_y)=0
$$
because the <Sobolev trace> of $v_h$ vanishes. Consequently
$$
\boxed{U^TPU=-\int_\Omega|\nabla v_h|^2\leq0.}
$$
Along the solution, $\int_\Omega u_h^2=U^TMU$. The constant <symmetric matrix> $M$ and the semidiscrete equations therefore give
$$
\boxed{\frac d{dt}\int_\Omega u_h^2=2U^TM\dot U=2U^TPU=-2\int_\Omega|\nabla u_h|^2\leq0.}
$$
The <L2 norm> of the spatial finite element solution is nonincreasing. The constant drift transports it without changing this energy; the diffusion term dissipates it. Any subsequent <time-stepping method> must be chosen to preserve an appropriate discrete stability estimate.
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