A Modified Weak Galerkin Finite Element Method for Singularly Perturbed Parabolic Convection-Diffusion-Reaction Problems (Journal of Computational Mathematics, 2023)
In this work, a modified weak Galerkin finite element method is proposed for solvingsecond order linear parabolic singularly perturbed convection-diffusion equations. Thekey feature of the proposed method is to replace the classical gradient and divergenceoperators by the modified weak gradient and modified divergence operators, respectively.We apply the backward finite difference method in time and the modified weak Galerkinfinite element method in space on uniform mesh. The stability analyses are presentedfor both semi-discrete and fully-discrete modified weak Galerkin finite element methods.Optimal order of convergences are obtained in suitable norms. We have achieved the sameaccuracy with the weak Galerkin method while the degrees of freedom are reduced in ourmethod. Various numerical examples are presented to support the theoretical results. Itis theoretically and numerically shown that the method is quite stable.
Keywords: The modified weak Galerkin finite element method; Backward Euler method; Parabolic convection-diffusion problems; Error estimates.