小参数对流扩散方程在最优分层网格的一致收敛有限元计算

Uniformly convergent finite element computation on optimally graded meshes for the convection-diffusion equation with a small parameter

面向小参数的奇异摄动对流扩散方程,构建分层网格自适应地刻画边界层对应的离散结点,应用有限元计算以期在特殊网格上得到优化结果.分层网格无需复杂计算,仅根据递推关系可形成随机剖分数的优化网格,实现更好地捕捉边界层.数值算例验证了方法的鲁棒性,获得了完全独立于小摄动参数、一致收敛的有限元高精度数值结果.

As for a singularly perturbed convection-diffusion equation with a small parameter,a graded mesh is built adaptively to describe the discrete nodes of the boundary layers,and the finite element computation is applied to solve the optimal results on special meshes.The graded mesh is free of complicated operations,it can be realized from a recursive formulation to build the optimal mesh with random partition numbers,which is qualified to capture the boundary layers.The robustness of the method is shown through numerical experiments,the high precision numerical results of finite element are obtained,which are completely independent of small perturbation parameters and uniformly convergent.