Galerkin method with discontinuous basis functions on staggered grips a priory estimates for the homogeneous Dirichlet problem

In this paper we present the accuracy of solution a priory estimates of a homogeneous boundary value problem for a second-order differential equation by the Galerkin method with discontinuous basis functions on staggered grids. To approximate the initial elliptic equation with known initial boundary conditions by the Galerkin method with discontinuous basis functions, it is necessary to transform it to a system of first-order partial differential equations. To do this auxiliary variables, representing the components on the flux of the sought value, are introduced. The characteristic feature of the method is the finding of auxiliary variables on the dual grid cells. The dual grid consists of median reference volumes and is conjugate to the basic unstructured triangular grid. The numerical fluxes on the boundary between the elements are found by using stabilizing additives. We show that for the stabilization parameter of order one, the $L^2$-norm of the solution is of order $k+\frac{1}{2}$, if the 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.