I am solving the steady, two-dimensional adjoint Euler equations, $$A_x^T \partial_x \Psi + A_y^T \partial_y \Psi = 0$$, where $A_x = \partial F_x/\partial U$ and $A_y= \partial F_y/\partial U$ are the Flux Jacobians and $\Psi = (\psi_1,\psi_2,\psi_3,\psi_4)^T$ is the adjoint state. $F_x=(\rho v_x,\rho v_x^2+p,\rho v_x v_y, \rho v_x H)^T$ and $F_y=(\rho v_y,\rho v_x v_y, \rho v_y^2+p, \rho v_y H)^T$. For external flows (flow around an airfoil, for example) $\Psi$ obeys dual characteristic b.c. at the far-field (whereby the outgoing characteristic components of the adjoint field are set to zero), while at solid walls the b.c. is of the form $(\psi_2,\psi_3)\cdot (n_x,n_y)=f(p)$ where $(n_x,n_y)$ is the normal vector to the surface and $f(p)$ is a function that depends on the choice of cost function. This type of problem is relevant for the computation of sensititivities of the cost function with respect to perturbations of the flow.
For practical applications, the above equations are solved on a computational mesh with (typically) a finite-volume discretization. Now it turns out that, for various solvers and depending on the choice of flow regime and cost function, the adjoint solution is strongly mesh-dependent near solid walls, i.e., the values of the adjoint fields at and near the wall grow continually as the mesh is refined. Besides, such mesh-dependence seems to be strongly correlated with the presence of singularities at the forward stagnation streamline and at the trailing edge / rear stagnation point [see https://www.eucass.eu/doi/EUCASS2019-0291.pdf ]. I would like to know if someone has experienced the same or a similar phenomenon.