Uniform convergent modified weak Galerkin method for convection-dominated two-point boundary value problems

OpenAlex İngilizce

We propose and analyze a modified weak Galerkin finite element method (MWG-FEM) for solving singularly perturbed problems of convection-dominated type. The proposed method is constructed over piecewise polynomials of degree $k\geq1$ on interior of each element and piecewise constant on the boundary of each element. The present method is parameter-free and has less degrees of freedom compared to the classical weak Galerkin finite element method. The method is shown uniformly convergent for small perturbation parameters. An uniform convergence rate of $\mathcal {O}((N^{-1}\ln N)^k)$ in the energy-like norm is established on the piecewise uniform Shishkin mesh, where $N$ is the number of elements. Various numerical examples are presented to confirm the theoretical results. Moreover, we numerically confirm that the proposed method has the optimal order error estimates of $\mathcal {O}(N^{-(k+1)})$ in a discrete $L^2$-norm and converges at superconvergence order of $\mathcal {O}((N^{-1}\ln N)^{2k})$ in the discrete $L^\infty$-norm.