In this dissertation , the numerical solution of singularly perturbed problems for a class of semilinear ordinary equations involving two parameters , a class of elliptic equations with turning points , a semilinear parabolic equation , and a two level semilinear parabolic equation as well as a hyperbolic - parabolic partial differential equation are mainly considered . the bounds on the derivatives of these exact solutions are given . then different difference schemes are constructed for different problems 本论文主要讨论了一类含双参数半线性常微分方程、一类具有转向点的椭圆型方程、半线性抛物型方程、二维半线性抛物型方程和双曲-抛物型偏方程奇摄动问题的数值解,首先给出这些方程解的导数估计,然后对不同问题构造了不同的差分格式,最后我们证明了该差分格式关于小参数一致收敛
A kind of operator of trigonometric interpolation polynomials with bivariate product was constructed based on an equidistant node set , so that the operator could be converged uniformly to bivariate continuous functions with periodicity of 27 on whole plane , and the convergence order of the approximation would be optimal for a body of functions with arbitrary - ordered continuous partial deritives 摘要构造了一类基于等距结点组上的二元三角插值多项式算子，使得该算子在全平面上一致收敛到每个以2为周期的二元连续函数，并且对具有任意阶连续偏导数的函数全体的逼近具有最佳收敛阶。
In chapter 1 , we use the p1 nonconforing mortar element to discrete poisson equation , and adopt multigrid method to solve the discrete equations , we have proven that the algorithm has a uniform convergence rate , that is , the convergence rate is independent of the mesh size and level . the multigrid method for the stationary stokes equations based on mortar mixed element method is presented in chapter 2 , and the uniform convergence rate of the method independent of mesh size and level for the w - cycle is proven 第二章求解的问题是定常stokes方程，用混合mortar元来离散原方程，每个子区域上用taylor - hood有限元进行离散，离散出的方程组用w -循环多重网格方法进行求解，第四节中证明了多重网格方法的一致收敛性，即收敛率与网格层数和网格尺寸无关。
