AI 中文总结
研究抛物型奇异摄动边界转折点问题，核心方法是用弱伽辽金有限元方法，在均匀网格时间离散与层适应Shishkin网格空间离散，建立稳定性与误差估计，验证理论结果，为高维问题扩展奠定基础。
AI 中文摘要
本文针对一类抛物型奇异摄动边界转折点问题（SPBTPPs）引入了一种弱伽辽金有限元方法（WG-FEM）。所提出的数值格式在均匀网格上采用隐式θ格式进行时间离散，并在层适应的Shishkin网格上应用WG-FEM空间离散。对半离散和全离散格式建立了严格的稳定性估计。此外，推导了能量范数下的误差估计，并证明了该格式的收敛性关于摄动参数是一致的。进行了数值试验以验证理论结果并说明所提方法的有效性。本文建立的理论框架为未来使用ADI型算子分裂WG-FEM格式扩展到高维问题奠定了基础。
英文摘要
In this article, we introduce a weak Galerkin finite element method (WG-FEM) for a class of parabolic singularly perturbed boundary turning point problems (SPBTPPs). The proposed numerical scheme employs an implicit $θ$-scheme for temporal discretization over a uniform mesh and applies WG-FEM spatial discretization on a layer-adapted Shishkin mesh. Rigorous stability estimates are established for both the semi-discrete and fully-discrete formulations. Furthermore, we derive error estimates in the energy norm and prove that the convergence of the scheme is uniform with respect to the perturbation parameter. Numerical tests are conducted to verify the theoretical findings and illustrate the efficiency of the proposed method. In addition, the theoretical framework developed in this work lays the foundation for future extensions to higher-dimensional problems using ADI-type operator splitting WG-FEM schemes.
