C1连续型广义有限元格式

C$^{1}$连续,即一阶导数连续.C$^{1}$连续型插值格式具有同时适用于离散PDE的弱形式与强形式的优点--即一种插值格式可以在使用PDE弱形式还是强形式之间做出选择,从而构造出更加高效的数值方法.由于单位分解广义有限元方法 (PUFEM, Babu${\check{ s}}$ka andMelenk(1997)),允许用户根据局部解的特征自定义任意高阶局部近似,具有精度高、程序实现与传统有限元相容性好的特点而受到广泛关注.但是,其总体近似函数的光滑性是由其所采用的单位分解函数--一般为标准有限元形函数--的光滑性所决定,因此多为C$^{0}$连续.如何在C$^{0}$连续标准有限元形函数的基础上,构造出满足C$^{1}$连续的总体近似函数,是一个仍未解决的问题.本文在作者前期研究的无额外自由度的单位分解插值格式的基础上,仅基于C$^{0}$标准有限元形函数,构造出至少C$^{1}$连续的无额外自由度单位分解格式.针对Poisson方程,讨论了该格式对PDE弱形式与强形式的离散.测试结果表明,方法可以同时用于弱形式与强形式的数值求解,而且可以在不改变网格和自由度数的前提下,获得高阶收敛.使用该插值格式的条件是:网格须是直角坐标网格(不要求均匀).该插值格式可以同时用于流体力学问题和使用欧拉背景网格求解动量方程的固体力学方法,如材料物质点法(materialpoint method).对于强形式的欧拉网格求解,该插值格式与"差分"不同之处在于,它具有有限元一样的在任意点处进行"插值"的特点.对于弱形式的积分求解,由于该插值格式具有导数连续性,可以允许积分网格独立于插值网格.这一特点将使得弱形式的数值积分的实施更加灵活方便. 

A GFEM WITH C$^1$ CONTINUITY

The function $f$ is said to be of class C$^{1}$ if the first order derivatives of $f$ exist and are continuous. A C$^{1}$ approximate can be applicable, totally up to users' choice, to solve the weak or the strong forms of PDEs, which provides an opportunity on designing a better-fit numerical method. The Partition of Unity Finite Element Method (PUFEM, Babuska and Melenk (1997)) gains broad attention due to a strikingly advantageous feature: A user-tailorablly high order approximation while without complicating numerical implementations in a standard FE code. However, the smoothness of the global approximate function of PUFEM is inherent to that of the partition of unity function that is usually taken as the standard finite element shape function. How to construct a PUFEM of class C$^{1}$ based on the C$^{0}$ finite element shape functions is still a pending problem. Based on the recently developed extra-dof free partition of unity approximation, we develop in this paper a C$^{1}$ continuous generalized finite element approximation using only a C$^{0}$ finite element shape function constructed on a Cartesian grid. The approximation is applied to discretize the Poisson's equation in both strong forms and weak forms. Numerical tests show that the approximation can be applicable to numerical solution to both the strong and the weak form of PDEs, and it is able to deliver a high order of accuracy and convergence without necessarily altering grid topology and increasing nodes. The necessary condition for using the C$^{1}$ approximate is a domain discretization based on a Cartesian grid (not needed to be uniform). The new approximate can be used to both fluids (like FDM) and solids (like material point method). The difference from FDM is that the field function and its derivatives at an arbitrary point of the domain of interest can be computed directly using the "shape function" of the new approximate. When working with the material point method, the new approximate can be expected to reduce or eliminate quadrature errors and cross-grid oscillations.