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Reproducing kernel element method. II: Globally conforming $$I^m/C^n$$ hierarchies. (English) Zbl 1093.74062
Summary: In this part of the work [for part I see the first four authors and J. Cao, ibid. 193, No. 12–14, 933–951 (2004; Zbl 1060.74670)], a minimal degrees of freedom, arbitrary smooth, globally compatible, $$I^m/C^n$$ interpolation hierarchy is constructed in the framework of reproducing kernel element method (RKEM) for arbitrary multiple dimensional domains. This is the first interpolation hierarchical structure that has been constructed with both minimal degrees of freedom and higher-order smoothness or continuity over multi-dimensional domain. The proposed hierarchical structure possesses the generalized Kronecker property, i.e.
$\partial^{\alpha} \Psi_I^{(\beta)}/\partial x^{\alpha} (x_j) = \delta_{IJ}\delta_{\alpha\beta}, \quad |\alpha|,|\beta| \leq m.$
This contribution is the latest breakthrough of an outstanding problem – construction of a minimal degrees of freedom, globally conforming, $$I^m/C^n$$ finite element interpolation fields on an arbitrary mesh or subdivision of multiple dimension. The newly constructed globally conforming interpolant is a hybrid of a set of $$C^{\infty}$$ global partition polynomials with a highly smooth ($$C^n$$) 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.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74S05 Finite element methods applied to problems in solid mechanics
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