The Superconvergence Of Discontinuous Galerkin Methods For Elliptic Equations

The discontinuous Galerkin method for various problems has been one of the highlights in the study of numerical methods, and has been applied to the fields of science and engineering widely in recent years. In this paper, the discontinuous Galerkin method for an elliptic equation is studied. The supercon-vergence of md-LDG method has been proved. Actually, for md-LDG method, the leading terms of the discretization errors for the DG solutions U and Q are proportional to the right Radau and left Radau polynomials of degree p + 1 on each element, respectively. This fact implies that the p + 1—degree right Radau points and left Radau points are the p + 2—degree superconvergence points of U and Q, respectively. Both the uniform and nommiform meshes are used to solve the elliptic equation. Numerical experiments validate our theoretical findings.On the other hand, motivated by the continuous Galerkin finite element method(CG), we combine the classical DG method and propose a new DG method originally. At the boundary points, the boundary conditions are imposed as it is done in CG. Nevertheless, in the inner elements of the domain, the DG method is implemented. Under the uniform mesh, two Shishkin-type meshes and two A—graded meshes, many numerical experiments are performed. These numerical results demonstrate that this new DG approach has perfect convergence and superconvergence consequences, which are as good as those obtained by the classical DG method. It is worthwhile to point out the new approach is simpler than the classical DG method, and its computational cost is lower.