A priori error estimates of the local discontinuous Galerkin method on staggered grids for solving a parabolic equation for the homogeneous Dirichlet problem

Abstract

In this paper, we present a priori error analysis of the solution of a homogeneous boundary value problem for a second-order differential equation by the Discontinuous Galerkin method on staggered grids. The spatial discretization is constructed using an appeal to a mixed finite element formulation. Second-order derivatives cannot be directly matched in a weak variational formulation using the space of discontinuous functions. For lower the order, the components of the flow vector are considered as auxiliary variables of the desired second-order equation. The approximation is based on staggered grids. The main grid consists of triangles, the dual grid consists of median control volumes around the nodes of the triangular grid. The approximation of the desired function is built on the cells of the main grid, while the approximation of auxiliary variables is built on the cells of the dual grid. To calculate the flows at the boundary between the elements, a stabilizing parameter is used. Moreover, the flow of the desired function does not depend on auxiliary functions, while the flow of auxiliary variables depends on the desired function. To solve this problem, the necessary lemmas are formulated and proved. As a result, the main theorem is formulated and proved, the result of which is a priori estimates for solving a parabolic equation using the discontinuous Galerkin method. The main role in the analysis of convergence is played by the estimate for the negative norm of the gradient. We show that for stabilization parameter of first order, the $L^2$-norm of the solution is of order $k+{1}/{2}$, if stabilization parameter of order $h^{-1}$ is taken, the order of convergence of the solution increases to $k+1$, when polynomials of total degree at least $k$ are used.

Keywords

• a priori error analysis
• finite element method
• discontinuous galerkin method
• staggered grids
• parabolic problems