We propose a novel finite element method scheme for singularly perturbed advection-diffusion-reaction problems, which combines certain quantum-assisted stabilization scheme with a classical h-adaptive approach to provide automatic error control and corresponding approximation refinement. Appropriate finite element a posteriori error estimates are proved. Described approach demonstrates the possibility to overcome singular perturbations by applying error-controlled smoothening to finite element approximation. Possible benefits of the proposed finite element scheme are discussed and the numerical comparison with the typical adaptive scheme is provided.

翻译：针对奇异摄动对流-扩散-反应问题，本文提出一种新型有限元方法方案。该方案将特定的量子辅助稳定化方案与经典的h自适应方法相结合，以实现自动误差控制及相应的近似解细化。文中证明了适当的有限元后验误差估计。所述方法通过对有限元近似施加误差控制的平滑处理，展示了克服奇异摄动的可能性。本文讨论了所提有限元方案的潜在优势，并提供了与典型自适应方案的数值比较。