A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

In this paper, we design and develop a weak Galerkin finite-element numerical method for solving singularly perturbed convection–diffusion–reaction equations with a non-conservation convection term. Many advantages of the proposed method include support for the higher order of convergence and general polygonal meshes. The convergence study for the weak Galerkin algorithm is performed in both the triple-bar norm and the \(L^2\) $L^2$ norm. We achieve an optimal order of convergence of \(\mathcal{O}(h^k)\) $\mathcal{O}(h^k)$ in the triple-bar norm and \(\mathcal{O}(h^{k+1})\) $\mathcal{O}(h^{k+1})$ in the \(L^2\) $L^2$ norm. Several numerical experiments in a two-dimensional setting are carried out to demonstrate the convergence of our theories.