抛物型方程有限元解最大Lp正则性及其在误差分析中的应用

This project will study maximal Lp-regularity of finite element solutions of parabolic partial differential equations (PDEs). Moreover, the errors of finite element solutions of parabolic PDEs will be analyzed by applying maximal Lp regularity. The advantage of maximal Lp-regularity is that it can derive the uniform boundedness of numerical solutions in some strong norms. By the uniform boundedness of numerical solutions, we can analyze the stability and accuracy of numerical solutions of nonlinear PDEs. Thus, we can study the errors of finite element solutions of strongly nonlinear and coupling equations by applying this technique. In this project, discrete maximal Lp-regularity will be applied to study the convergence and error estimates of finite element solutions of the three parabolic PDEs with strong nonlinearities and couplings: (1) We will study the finite element methods for the porous medium flow with only Lipschitz continuous coefficients. Optimal Lp(Lq) error estimates and almost optimal L∞(Lq) error estimates for fully discrete solutions will be established. (2) We will study the error estimates of finite element solutions of Navier-Stokes equations with variable density. (3) Mixed finite element methods for Cahn-Hilliard-Navier-Stokes equations will be studied in general polyhedral domain (possibly non-convex and multi-connected). The convergence analysis of numerical solutions will be provided without any assumptions on the regularity of solutions. Finally, numerical experiments will be given to verify theoretical analysis.

本项目研究了抛物型偏微分方程(PDEs)有限元解最大Lp正则性,并利用最大Lp正则性分析抛物型PDEs有限元解的误差.最大Lp正则性优点在于可以得到数值解在强范数下一致有界性.由数值解一致有界性就可以分析非线性PDEs数值解稳定性和精度.所以该技术可用来研究强非线性,强耦合性方程有限元解的误差.在本项目中利用离散最大Lp正则性研究了三类强非线性,强耦合性抛物型方程有限元解收敛性和误差估计:(1)研究系数仅Lipschitz连续多孔介质流的有限元法,得到了全离散解最优Lp(Lq)误差估计和几乎最优L∞(Lq)误差估计.(2)研究变密度N-S方程有限元解的误差估计.(3)研究在一般多面体区域(可能非凸,多连通)Cahn-Hilliard-Navier-Stokes方程混合元法,对解不做正则性假设情况下,给出数值解的收敛性分析.最后,数值实验将被给出以验证理论分析.

本项目研究了抛物型偏微分方程(PDEs)有限元解的最大Lp正则性,并利用最大Lp正则性分析了抛物型PDEs有限元解的误差.最大Lp正则性优点在于可以得到数值解在强范数下一致有界性.由数值解一致有界性就可以分析非线性PDEs数值解的稳定性和精度.所以该技术可用来研究强非线性,强耦合性方程有限元解的误差.本项目研究了一些强非线性,强耦合性抛物型方程有限元解的收敛性和误差估计:(1)研究了多孔介质流方程的Crank-Nicolson有限元方法,得到了全离散解最优误差估计.(2)研究了变密度的Navier-Stokes方程有限元解的误差估计.(3)证明了Cahn-Hilliard-Navier-Stokes方程的凸分裂有限元方法的最优L2误差估计.最后,数值实验被给出以验证理论分析结果.

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