$(1+s)$-Order Convergence Analysis of Weak Galerkin Finite Element Methods for Second Order Elliptic Equations (Advances in Applied Mathematics and Mechanics, 2020)
In this paper, we prove a lower convergence rate of the weak Galerkin finite element method for second order elliptic equations under a standard weak smoothness assumption. In all present literatures on weak Galerkin finite element methods for second order PDEs, the $H^2$ smoothness is compulsorily assumed for the real solution and hence a second order convergence is obtained. This lead to that the piecewise linear functions are excluded to construct finite element bases, although they behave very well in all numerical experiments. We intend to prove the $(1+s)$-order convergence rate under the $H^1$-smoothness assumption of the real solution and an additional $s>0$ regularity of the dual problem. Our strategy is that we firstly approximate the elliptic problem using the traditional finite element method with at least $H^2$ smooth bases, and then we apply the weak Galerkin method to approach this smooth approximating solution. Our result is an important supplementary for the weak Galerkin finite element method theory.