Difference interior penalty discontinuous Galerkin method for the 3D elliptic equation

Abstract

This paper presents a difference interior penalty discontinuous Galerkin method for the 3D elliptic boundary-value problem. The main idea of this method is to combine the finite difference discretization in the $z$-direction with the interior penalty discontinuous Galerkin discretization in the $(x,y)$-plane. One of the advantages of this method is that the solution of 3D problem is transformed into a series of 2D problems, thereby overcoming the computational complexity of traditional interior penalty discontinuous Galerkin method for solving high-dimensional problems and allowing for code reuse. Additionally, we use the interior penalty discontinuous Galerkin method to solve each 2D problem, therefore, this method retains the advantage of the interior penalty discontinuous Galerkin method in dealing with non-matching grids and non-uniform, even anisotropic, polynomial approximation degrees. Then, the error estimates are given for difference interior penalty discontinuous Galerkin method. Finally, numerical experiments demonstrate the accuracy and effectiveness of the difference interior penalty discontinuous Galerkin method.