In this paper, we design and analyze the iterative two-grid methods for the discontinuous Galerkin (DG) discretization of semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that is just like the classical iterative two-grid methods for nonsymmetric or indefinite linear elliptic PDEs, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Secondly, we designed another iterative two-grid method, which replace the semilinear term by using the corresponding first-order Taylor expansion. Specifically, we need to construct a suitable initial value, which can be sorted out from an auxiliary variational problem, for the second iterative method. We also provide the error estimates for the second iterative algorithm and present numerical experiments to illustrate the theoretical result.

在本文中，我们设计并分析了半线性椭圆偏微分方程（PDE）的不连续伽辽金（DG）离散化的迭代两网格方法。我们首先提出了一种迭代两网格方法，它与非对称或不定线性椭圆偏微分方程的经典迭代两网格方法类似，即先求解粗空间上的半线性问题，然后再求解粗空间上的对称正定问题。精细的空间。其次，我们设计了另一种迭代双网格方法，用相应的一阶泰勒展开式代替半线性项。具体来说，我们需要为第二次迭代方法构造一个合适的初始值，该初始值可以从辅助变分问题中整理出来。我们还提供了第二个迭代算法的误差估计，并提供了数值实验来说明理论结果。