PDE 7: 2D Stationary advection-diffusion equation

Given a solenoidal vector field \(\mathbf{V}\) (i.e. \(\nabla\cdot \mathbf{V}=0\) ) on the square \(0\leq x\leq 1 , 0\leq y\leq 1 \), we consider the equation

\[\nu \nabla^2 u + \mathbf{V}\cdot\nabla u = 0 \]

For boundary conditions take \(\Gamma=\Gamma_D\cup\Gamma_N\) where \[\Gamma_D=\{(x,0), 0\leq x\leq 1\}\cup \{(0,y), 0\leq y\leq 1\}, \] \[\Gamma_N=\{(x,1), 0\leq x\leq 1\}\cup \{(1,y), 0\leq y\leq 1\}. \]

We assume that \(u(x,y)=u^*=\mbox{konst}\) when \((x,y)\in\Gamma_D\) and that \(\nabla u\cdot\vec{n}=0\) when \((x,y)\in\Gamma_N \), where \(\vec{n}\) is the outward-facing normal. The course Matematikk 2 should have taught you how to construct divergence-free vector fields in \(\mathbf{R}^2\), but if you have problems please ask! Investigate the role of the parameter \(\nu\).

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