Preconditioned Nonsymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation

In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. This method is originally proposed in Li et al. (J Sci Comput 80(1):268–288, 2019) that an arbitrarily high-order approximation space with one unknown per element is reconstructed by solving a local least squares fitting problem. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin methods. The least squares problem is modified in this paper, which allows us to establish a norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further inspires us to construct a preconditioner from the piecewise constant space. The preconditioner is shown to be optimal that the upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is independent of the mesh size. In addition, we can enjoy the advantage on the efficiency of the approximation in number of degrees of freedom compared with the standard DG method. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method.