The linear convection-diffusion equation is the partial differential equation
-a\del^2 u +\del.(bu) =0
where the 2nd order term models diffusion and the first order term models convection. This equation is classically very difficult to solve numerically. Standard methods lead to excessive unrealistic oscillations when the magnitude of the convective field dominates the diffusion. The above graphic is a contour plot of the numerical solution obtained from a Petrov-Galerkin discretisation of the above equation in two dimensions with diffusion parameter a=0.1 and with a convective field of
b(x,y)=(2y(1-x^2),-2x(1-y^2))^T.
For this problem the inlet boundary condition along x in [-1,0] ,y=0 is given by U(x,0)=1+tanh[20x+10] . The boundary condition on the tangential boundaries, x=-1,y=1 and x=1 is given by U=0, and a homogeneous Neumann boundary condition is placed on the outflow boundary x>0,y=0.