本项目以奇异摄动问题为研究对象，拟对三角形及矩形层适应网格（包括Shishkin型网格、Bakhvalov型网格等）上几类有限元方法（包括标准有限元方法以及流线扩散方法、连续内部加罚方法和局部L^2投影方法等稳定化有限元方法）的超逼近性进行研究，并应用超逼近性构造后处理算子，进而得到超收敛结果。本项目中，三角形层适应网格上几类有限元方法（利用线性元）超逼近性的研究将改变现有分析局限在矩形层适应网格上的现状；三角形网格上高次元扩散项及对流项局部误差估计式的构造，可大大促进现有超收敛理论及自适应有限元方法的发展，包括协调元、非协调元、混合元在流体、磁场及材料等领域的应用；三角形及矩形层适应网格上几类有限元方法（利用高次元）超逼近性的研究将使得目前奇异摄动领域高阶数值方法的研究取得突破性进展。

In this project we aim at singularly perturbed problems, analyze supercloseness properties of several finite element methods (standard finite element methods, streamline diffusion finite element methods, continuous interior penalty methods and local L^2 projection stabilization methods etc.) on triangular and rectangular layer-adapted meshes (Shishkin type meshes, Bakhvalov type meshes etc.), construct postprocessing operators and obtain superconvergence results. In this project, we are going to analyze supercloseness properties of several finite element methods using linear elements on triangular layer-adapted meshes, which will improve researches on rectangular layer-adapted meshes. Also, we are going to construct local error estimation expressions for diffusion terms and convection terms with high-order finite elements on triangular meshes. These expressions can promote the superconvergence theory and adaptive finite element methods, which includes the applications of conforming finite elements, non-conforming finite elements and mixed finite elements in fluid, magnetic field and material and so on. Furthermore it will be of great development in the numerical theories of high-order methods for singularly perturbed problems to analyze supercloseness properties of several finite element methods with high-order finite elements on triangular and rectangular layer-adapted meshes.