I've been given an exercise which goal is to study the convergence of the finite elements method for an elliptic problem in the form $$ \begin{align} -\text{div}(k \nabla u) + \beta \cdot \nabla u + \alpha u & = f \qquad \text{in} \ \Omega \\[4pt] u & = g \qquad \text{on} \ \partial \Omega \end{align} $$
Since the exercise is let on purpose "free", we are suggested to choose $k, \beta, \alpha, f,$ the domain $\Omega \subseteq \mathbb{R^2}$ and also the exact solution $u$ of the problem.
Since the coefficients must satisfy the ellipticity of the natural bilinear map, we have the following costraints: $$ k(x,y) \ge k_0 > 0 \qquad -\frac{1}{2} \text{div}(\beta(x,y)) + \alpha(x,y) \ge 0$$ where $k_0$ is costant.
I would like some suggestions on theese choiches (hopefully not-trivial examples) that perhaps could lead to some conclusions better than others, and I am sure that your experience in this field will help me a lot! Thank you.