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.