几类分数阶偏微分方程的虚拟有限元方法

The virtual finite element method is considered to be a generalization of the finite element method to polygonal or polyhedral meshes. The stiffness matrix is realized only by the calculation of the degree of freedom, which no need to know the specific expression form of the shape function. Thus the realization of the program is simpler and the efficiency of computing is greatly improved. In this project, we mainly consider the virtual finite element methods of some types of fractional partial differential equations. In regard to the time fractional partial differential equations (containing the cases of smooth and non-smooth initial values), we consider the virtual finite element methods based on the higher order temporal discretize schemes, and then focus on the virtual element construction, stability, convergence and programming implementation. Specially, for the linearized virtual finite element methods of some types of the nonlinear time fractional models, the time-space error splitting technique is adopted for the analysis of the unconditional convergence properties. Moreover, in regard to the space fractional partial differential equations, this project is focused on the virtual element construction related to the model itself, the corresponding functional spaces are established, the analysis of the solvability, stability and convergence are carried out, and finally the implementation of the virtual finite element program based on the polygon or polyhedron meshes is further studied. This project will expand the application scopes of the virtual finite element methods, explore the general rules of solving the fractional order model numerically, and be expected to be applied and developed in more fields.

虚拟有限元方法为有限元方法在多边形或多面体网格上的推广，其在计算刚度矩阵时仅仅通过自由度的运算来实现，无需知道形函数的具体表达形式，从而该方法编程实现更加简单，计算效率得到显著提高。本项目考虑几类分数阶偏微分方程的虚拟有限元方法。对于时间分数阶偏微分方程（初值光滑与非光滑），本项目考虑其基于高阶时间离散格式的虚拟有限元方法，重点研究虚拟单元构造、稳定性、收敛性以及程序实现。特别地，针对几类非线性时间分数阶模型的线性化虚拟有限元方法，本项目拟采用时间-空间误差分裂技巧研究数值方法的无网格比收敛性质。另外，对于空间分数阶偏微分方程，本项目重点研究与模型本身相关的虚拟单元构造，建立对应函数空间，并进行适定性、稳定性及收敛性分析，最后进一步研究其基于多边形或多面体网格下的虚拟有限元程序实现。本项目将拓展虚拟有限元方法的应用范围，探索该方法数值求解分数阶模型的一般规律，以期在更多领域得以应用与发展。

虚拟元方法为有限元方法在多边形或多面体网格上的推广，可以称为多边形或者多面体有限元方法。该方法在计算刚度矩阵时仅仅通过自由度的运算来实现，无需知道形函数的具体表达形式，从而该方法编程实现更加简单，计算效率得到显著提高。本项目主要考虑几类分数阶偏微分方程的虚拟有限元方法, 另外我们也考虑了一些非线性模型的虚拟元方法。对于二阶及四阶时间分数阶偏微分方程（初值光滑与非光滑），本项目考虑其基于高阶时间离散格式的虚拟有限元方法，重点研究虚拟单元构造、稳定性、收敛性以及程序实现。另外，对于非线性薛定谔模型，我们采用不同的方法研究了该模型协调及非协调虚拟元方法的适定性、稳定性及收敛性等。本项目拓展了虚拟元方法的应用范围，探索了该方法数值求解偏微分模型的一般规律，在更多领域得以应用与发展。

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