Discontinuous Galerkin schemes with defect corrections based on reconstruction

Reconstruction is usually thought of a building block in finite volume schemes, but it has also recently been combined with discontinuous Galerkin (DG) schemes. In this approach, the degrees of freedom of a piecewise polynomial approximation of degree N are directly based on the DG variational formulation, while reconstruction is used to raise the polynomial degree of the approximation to M ⩾ N and thus increase the order of accuracy of the final solution. In this paper, we propose the use of reconstruction to estimate the local discretization error of a steady state solution of a discontinuous Galerkin scheme. An iterated defect correction is then applied to improve the accuracy of the steady solution, by placing the estimate for the error on the right hand side. Hence, within this approach one only needs the inversion of the basic lower-order DG scheme. The main advantage is that the defect correction does not affect the DG scheme beside a modification of the right hand side, and the matrix of the linear system to be solved remains unchanged. For problems in which the computational effort for higher order schemes strongly increases with the order, then the defect correction scheme proposed here may be considerably more efficient. Numerical results for Euler and Navier–Stokes equations are shown.