In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. We reconstruct a high-order piecewise polynomial space that arbitrary order can be achieved with one degree of freedom per element. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin method. Compared with the standard DG method, we can enjoy the advantage on the efficiency of the approximation. Besides, we establish an norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further allows us to construct an optimal preconditioner from the piecewise constant space. The upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is shown to be independent of the mesh size. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method.

翻译：本文提出并分析了一种基于重构非连续逼近方法的椭圆问题高效预处理技术。我们重构了一个高阶分片多项式空间，该空间每个单元仅需一个自由度即可达到任意阶精度。该空间可直接与对称/非对称内部惩罚间断伽辽金方法配合使用。相较于标准DG方法，我们在逼近效率上具有显著优势。此外，我们建立了重构高阶空间与分片常数空间之间的范数等价关系，该性质进一步允许我们从分片常数空间构造最优预处理器。预处理后对称/非对称系统的条件数上界被证明与网格尺寸无关。数值实验验证了理论的有效性及所提方法的高效性。