三角形网格上Lagrange型高阶元有限体积法的稳定性、L^2误差估计和超收敛

Based on the model of second order elliptic equations, this project will study the stability, L^2 error estimates and superconvergence of Lagrange high-order finite element methods on triangular meshes...1) L^2 error estimates of high-order odd-order schemes. We have presented the L^2 error estimates for arbitrary order finite volume schemes which satisfy orthogonal conditions. However, with numerical experiments, we find that there are still many other high-order odd-order schemes which have optimal convergence rate with L^2 norm. The L^2 error estimates of these schemes can not be explained by orthogonal conditions. This project will complete the L^2 error estimates of high-order odd-order schemes which can not be covered by orthogonal conditions...2) Superconvergence of high-order schemes. We have already obtained the superconvergence of quadratic scheme. However, there exists no result about the superconvergence of cubic and higher order schemes. This project will study and carry out the theory of superconvergence for cubic and higher order schemes...3) Improve the stability of cubic scheme. There is a limit to the lower bound of the triangular mesh in the existing results about the stability of cubic scheme. This project will improve the restrict for the stability of cubic finite volume scheme on triangular meshes, and we hope to prove the cubic scheme is unconditionally stable.

本项目拟以二阶椭圆型方程为模型，研究三角形网格上Lagrange型高阶元有限体积法的稳定性、L^2误差估计和超收敛。. .1）高阶奇次元的L^2误差估计：我们已经对任意次满足正交条件的有限体积格式给出了L^2误差估计的证明。然而在数值实验过程中，我们发现了大量不满足正交条件的高阶奇次元有限体积格式同样具有最佳按L^2模收敛阶。本项目将补充解决正交条件所不能覆盖的有限体积格式的最佳L^2模收敛阶的证明。..2）高阶元的超收敛：目前我们已经对二次元的超收敛给出回答，而三次及更高次元尚无超收敛结果。本项目将对三次元及更高次元有限体积法的超收敛进行研究，给出三次元及更高次元有限体积法具有超收敛的条件，并加以证明。..3）三次元的稳定性：现有的三次元有限体积法的稳定性结果对三角形网格有最小角下界的限制。本项目将改善改进三次元有限体积法的稳定性对三角形网格的约束，最终证明格式无条件稳定。

项目的背景.近年来，有限体积法的理论和格式研究有了长足的发展，然而，对于有限体积法的稳定性、误差估计和超收敛的理论仍然才有很多需要发展的。..主要研究内容.目前已经完成的工作包括：三角形网格上有限体积法二次元有限体积法的超收敛结果、任意维单纯形上线性有限体积法的条件数分析；以及一维任意次有限体积格式具有最佳L2模收敛阶的充分必要条件、和有限体积法所特有的超收敛结构“正交超收敛结构”。..重要结果关键数据及其科学意义.主要结果有3个方面：.1.对于空间中任意维单纯形上线性元有限体积法的条件数作了系统的估计; 这对于有限体积法的计算效率有着积极的意义；2.对于一维给出了有限体积法具有最佳L2收敛阶的充分必要条件，这其中格式构造的自由性为解决一些类似于对流占优问题等复杂问题成为了可能；3.我们发现了有限体积法所特有的超收敛性质，即通过调整对偶剖分，超收敛点可以根据需要进行任意选取，这为提高有限体积法的计算效率，也为有限体积法满足某些特别的需求成为了可能。