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In this article, a new weak Galerkin finite element method is introduced to solve convection-diffusion-reaction equations in the convection dominated regime. Our method is highly flexible by allowing the use of discontinuous approximating functions on polytopal mesh without imposing extra conditions on the convection coefficient. An error estimate is devised in a suitable norm. Numerical examples are provided to confirm theoretical findings and efficiency of the method.

In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of Babuška et al., especially, we confirm that the computed data have 9 digits of accuracy with an exception of one pair (which has 8-7 digits of accuracy). In addition, we demonstrate that the function value superconvergent points predicted by the symmetry theory of Schatz et al. are the only superconvergent points for the Poisson equation. Finally, we provide function value superconvergent points for the Laplace equation, which are not reported elsewhere in the literature.

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