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Proof of linear independence of flat-top PU-based high-order approximation. (English) Zbl 1297.65141

Summary: This paper extends a rank deficiency counting approach, which was initially established by X. M. An et al. [Comput. Methods Appl. Mech. Eng. 200, No. 5–8, 665–674 (2011; Zbl 1225.74077); Comput. Methods Appl. Mech. Eng. 233–236, 137–151 (2012; Zbl 1253.65183)] to determine the rank deficiency of finite element partition of unity (PU)-based approximations, to explicitly prove the linear independence of the flat-top PU-based high-order polynomial approximation. The study also examines the coupled flat-top PU and finite element PU-based approximation, and the results indicate that the space at a global level is also linearly independent for 1-D setting and 2-D setting with triangular mesh, but not so for rectangular mesh. Moreover, a new procedure is proposed to simplify the construction of flat-top PU, and its feasibility, accuracy and efficiency have been validated by a typical numerical example.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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