无界区域椭圆型和抛物型偏微分方程的人工边界条件数值方法研究

项目名称： 无界区域椭圆型和抛物型偏微分方程的人工边界条件数值方法研究

项目编号： No.11471019

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 郑权

作者单位： 北方工业大学

项目金额： 65万元

中文摘要： 本项目开展椭圆型和抛物型偏微分方程外问题的人工边界条件数值方法的性质和应用研究。1)研究各向异性椭圆型偏微分方程外问题在非均匀网格椭圆、椭球人工边界上的自然边界归化法，得到有限元解依赖于超奇异积分核级数截断项数、人工边界大小、离散尺寸和准确解的先验误差估计；将结果推广到拟线性椭圆型偏微分方程外问题；并研究其h-自适应和r-自适应方法及其结合。2)研究Helmholtz方程外问题在非均匀网格圆、椭圆人工边界上的自然边界归化法，得到其离散解依赖于超奇异积分核的级数截断项数、人工边界大小、空间离散尺寸和准确解的先验误差估计；并研究其h、r-自适应法。3)研究抛物型偏微分方程外问题的整体人工边界条件差分法，得到该方法的稳定性和误差估计；研究利用非均匀网格和移动网格方法。4)研究非线性Burgers方程外问题的非线性人工边界条件差分法，得到该方法的稳定性和误差估计；研究利用非均匀网格和移动网格方法。

中文关键词： 计算数学；偏微分方程数值解；高精度数值方法；有限元方法；科学计算

英文摘要： This project investigates properties and applications of the numerical methods with artificial boundary conditions for the elliptic and the parabolic partial differential equations on unbounded domains. 1) Study the numerical method of the natural boundary reduction with non-uniform meshes on artificial ellipses/ellipsoids for solving the exterior problems of anisotropic elliptic partial differential equations, obtain the a priori error estimate of the finite element solution which depends on the item number of the truncated hypersingular integral kernel series, the mesh size of descretization, the width of the artificial boundary, and the exact solution; generalize the result to the exterior problems of quasilinear elliptic partial differential equations; and discuss the proposed method by using h-adaptive, r-adaptive and hr-adaptive techniques. 2) Study the numerical method of the natural boundary reduction with non-uniform meshes on artificial circles/ellipses for solving the exterior problems of Helmholtz equation, obtain the a priori error estimate of the finite element solution which depends on the item number of the truncated hypersingular integral kernel series, the mesh size of the descretization, the width of the artificial boundary, and the exact solution; and discuss the proposed method by using h-adaptive, r-adaptive and hr-adaptive techniques. 3) Study the finite difference method for solving the exterior problems of parabolic partial differential equations with nonlocal artificial boundary conditions, obtain the stability and the error estimate of the finite difference method, and discuss the method with non-uniform meshes and the moving mesh method. 4) Study the finite difference method for solving the exterior problem of the nonlinear Burgers' equation with the nonlinear artificial boundary condition, obtain the stability and the error estimate of the finite difference method, and discuss the method with non-uniform meshes and the moving mesh method.

英文关键词： Computational mathematics;Numerical solutions of PDE;Numerical method of high precision;Finite element method;Scientific computing