Numerical solutions to the adjoint Euler equations have been found to diverge with mesh refinement near walls for a variety of flow conditions and geometry configurations. The issue is reviewed and an explanation is provided by comparing a numerical incompressible adjoint solution with an analytic adjoint solution, showing that the anomaly observed in numerical computations is caused by a divergence of the analytic solution at the wall. The singularity causing this divergence is of the same type as the well-known singularity along the incoming stagnation streamline and both originate at the adjoint singularity at the trailing edge. The argument is extended to cover the fully compressible case, in subcritical flow conditions, by presenting an analytic solution that follows the same structure as the incompressible one.

翻译：数值求解伴随欧拉方程时，发现对于多种流动条件和几何构型，其解在壁面附近随网格细化而发散。本文回顾了这一问题，并通过对比数值不可压缩伴随解与解析伴随解，论证了数值计算中观测到的异常现象是由壁面处解析解的发散所导致的。导致这种发散的奇异性与沿来流驻点流线的已知奇异性属于同一类型，两者均源于后缘处的伴随奇异性。通过给出一个结构与不可压缩情形相同的解析解，将论证扩展至涵盖亚临界流动条件下的完全可压缩情形。