A Conforming Discontinuous Galerkin Finite Element Method for Second-Order Parabolic Equation

Abstract

The conforming discontinuous Galerkin (CDG) finite element method is an innovative and effective numerical approach to solve partial differential equations. The CDG method is based on the weak Galerkin (WG) finite element method, and removes the stabilizer in the numerical scheme. And the CDG method uses the average of the interior function to replace the value of the boundary function in the standard WG method. The integration by parts is used to construct the discrete weak gradient operator in the CDG method. This paper uses the CDG method to solve the parabolic equation. Firstly, the semi-discrete and full-discrete numerical schemes of the parabolic equation and the well-posedness of the numerical methods are presented. Then, the corresponding error equations for both numerical schemes are established, and the optimal order error estimates of $H^1$ and $L^2$ are provided, respectively. Finally, the numerical results of the CDG method are verified.