Error Analysis of a Fully-Discrete Implicit $\theta$-Scheme with WG-FEM for Parabolic Singularly Perturbed Boundary Turning Point Problems

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 $\theta$-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.