界面问题自适应高阶间断伽略金方法研究

界面问题在科学和工程领域都有着十分广泛的应用，如何对它设计高效高精度的数值方法是当今计算数学和工程领域的难点问题。本项目旨在基于非贴体网格，分别研究三维椭圆界面问题和H(curl)界面问题自适应高阶间断伽略金方法，并证明数值方法的稳定性，给出数值方法近似解的先验误差估计分析；结合超收敛重构技术，构造重构型后验误差估计子来指导网格自适应加密，设计相应的自适应算法。本项目的成功实施对完善界面问题的数值理论分析，推动多相流和波传播模拟具有重要意义。

Interface problems are widely used in science and engineering. How to design efficient and high-precision numerical methods for interface problems is a difficult problem in the field of computational mathematics and engineering. This project aims to study the adaptive high order discontinuous Galerkin method for three-dimensional elliptic interface problem and H(curl) interface problem, and prove the stability of the numerical method. The prior error estimation analysis of approximate solution of numerical method isinvestigated. By the superconvergence recovery techniques, recovery type a posteriori error estimators are designed to guide the mesh adaptive refinement and the corresponding adaptive algorithms are designed. The successful implementation of this project is of great significance for improving the numerical theory analysis of interface problems and promoting the simulation of multiphase flow and wave propagation.