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：本文针对基于重构间断逼近方法的椭圆问题，提出并分析了一种高效的预处理方法。我们重构了一个高阶分片多项式空间，该空间可在每个单元仅有一个自由度的条件下实现任意阶精度。此空间可直接与对称/非对称内部惩罚间断伽辽金方法结合使用。与标准DG方法相比，该方法在逼近效率方面具有显著优势。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。这一性质进一步允许我们从分片常数空间构造出最优预处理子。理论分析表明，预处理后对称/非对称系统的条件数上界与网格尺寸无关。数值实验验证了理论的有效性及所提方法的高效性。