Reproducing kernel element interpolation: Globally conforming Im/Cn/Pk Hierarchies

Abstract

In this work, arbitrarily smooth, globally compatible, Im/Cn/Pk interpolation hierarchies are constructed in the framework of reproducing kernel element method (RKEM) for multi-dimensional domains. This is the first interpolation hierarchical structure that has been ever constructed with both minimal degrees of freedom and higher order continuity and reproducing conditions over multi-dimensional domains. The proposed hierarchical structure possesses the generalized Kronecker property, i.e., ∂αΨI(β)/∂xα(xJ) = δIJδαβ, {pipe}α{pipe}, {pipe}β{pipe} ≤ m. The newly constructed globally conforming interpolant is a hybrid of global partition polynomials (C∞) and a smooth (Cn) compactly supported meshfree partition of unity. Examples of compatible RKEM hierarchical interpolations are illustrated, and they are used in a Galerkin procedure to solve differential equations.