 Solution

Use a conforming finite element space $V_h\subset H_0^1(\Omega )$ with the prescribed basis, and write $u_h(x,y,t)={\sum }_j{U}_j(t){\varphi }_j(x,y)$. Testing the advection-diffusion equation by ${\varphi }_i$ and applying integration by parts to its diffusion term givesThusHere $M$ 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 $v_h={\sum }_j{U}_j{\varphi }_j$. The skew symmetry of Galerkin advection with constant drift $(1,1)$ givesbecause the Sobolev trace of $v_h$ vanishes. ConsequentlyAlong the solution, ${\int }_{\Omega }{u}_h^2=U^{T}MU$. The constant symmetric matrix $M$ and the semidiscrete equations therefore giveThe 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.