Can you all give me (at least) one example about convection-dominated problem in order that I can test it (them) on FreeFEM++. If possible, please give me specific examples (it/they contain(s) full equations, boundary condition, initial condition, value of parameters, code in FreeFEM++,...)
Need an example of convection-dominated problem to test on FreeFEM++
finite element;numerical analysis
For $-\epsilon \Delta u + \mathbf{v}\cdot\nabla u = f$, where $\epsilon \ll |\mathbf{v}|$. You can refer to this paper:Adaptive anisotropic meshing for steady convection-dominated problems, section 6 and section 7.Computational Methods for Singularly Perturbed Systems, example 5.1. Known solution, boundary layers. This is also included in the NIST benchmark list.For the all diffusion-convection-reaction equation For $-\nabla\cdot(A\nabla u) + \mathbf{b}\cdot\nabla u + cu= f$, you can refer to A posteriori energy-norm error estimates for advection-diffusion equations approximated by weighted interior penalty methods, section 4.2, the right hand side $f$ is computed using the true solution.When the second order term is totally gone  $  \mathbf{b}\cdot\nabla u + \mu u= f$ (meaning the solution is less and less smooth and the problem becomes harder), there are two examples I know:http://hpfem.org/hermes-doc/hermes-examples/html/src/hermes2d/benchmarks-general/stabilized-advection-reaction.htmlA Posteriori Error Estimation for Interior Penalty Finite Element Approximations of the Advection-Reaction Equation, section 5. Since this is behind a paywall, I am list it here:Let $\Omega = \{(x,y)\in (0,1)\times (0,1)\}$. $\Omega^-$ is the left edge of the square. True solution $u$ satisfies$$\left\{ \begin{aligned}\mu u + \mathbf{b}\cdot \nabla u &= f,\\u|_{\partial \Omega^-} &=  g,\end{aligned} \right.$$with$$\mathbf{b} = \binom{y+1}{-x}\frac{1}{\sqrt{x^2 + (y+1)^2}}, \quad  \mu= 0.1$$and $g(x,y)$ chosen so that the exact solution is:$$u(x,y) = e^{\mu \sqrt{x^2 + (y+1)^2} \arcsin\left(\frac{y+1}{\sqrt{x^2 + (y+1)^2}}\right)} \arctan\left(\frac{\sqrt{x^2 + (y+1)^2} - 1.5}{\epsilon}\right).$$