Reproducing kernel element method. Part I: Theoretical formulation Original Research Article

In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of the shape functions. These properties are desirable especially in solving Galerkin weak forms of higher order partial differential equations and in treating Dirichlet boundary conditions. So unlike the FEM, there is no need for special treatment with the RKEM in solving high order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.