项目考虑高波数Helmholtz问题的数值算法的理论分析，包括PPR（Polynomial Preserving Recovery）方法和弱有限元算法（weak Galerkin finite element methods）。PPR方法是由张智民和Naga于2004年提出超收敛后处理方法，并已成功地被COMSOL Multiphysics采用。弱有限元方法是由王军平于2013年提出的求解偏微分方程的全新而有效的算法。本项目拟解决如下问题：1）基于PPR方法的Robin边界Helmholtz问题的数值解（连续有限元解和间断有限元解）的超收敛性估计，特别是PPR算法对污染误差的影响；2）改进已有的高波数Helmholtz问题的弱有限元解的误差估计。本项目的目标是探寻高波数Helmholtz问题的高效算法， 这些问题的研究在理论和实际应用上都有重要意义。

We consider the theoretical analysis of numerical methods for the Helmholtz equation with large wave number, which include the polynomial preserving recovery (PPR) method and the weak finite element methods. The PPR was proposed by Zhang and Naga in 2004 and has been successfully adopted by COMSOL Multiphysics as a post-proceeding tool. The weak finite element methods was proposed by Wang and Ye in 2013 as nonconforming methods for second-order elliptic euqations. We aim to achieve the following targets: 1) superconvergence analysis of FEMs (including DG FEM) based on the PPR method and the most important issue: Does PPR reduces the pollution error?; 2) improve the error estimates of weak Galerkin finite element methods. Our goal is to find efficient methods for the Helmholtz problem with large wave number, which has important theoretical significance and practical significance.