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Unformatted text preview: ere are no errors in the solution subspace. Bank 10] chose S N as a space of
discontinuous polynomials of the same degree p used for the solution space SE however,
the algebraic system for E resulting from (7.4.8) or (7.4.9) could be ill-conditioned when
the basis is nearly continuous. A better alternative is to select S N as a space of piecewise
p + 1 st-degree polynomials when SE is a space of p th degree polynomials. Hierarchical
bases (cf. Sections 2.5 and 4.4) are the most e cient to use in this regard. Let us
illustrate the procedure by constructing error estimates for a piecewise bilinear solution
on a mesh of quadrilateral elements. The bilinear shape functions for a canonical 2 2
square element are Ni1j (
where ) = Ni ( )Nj ( ) N1 ( ) = 1 ;
2 i j=1 2 N2 ( ) = 1 + :
(7.4.10b) The four second-order hierarchical shape functions are N32 j ( j=1 2 (7.4.11a) Ni23(
where ) = Nj ( )N32 ( )
) = Ni( )N32( ) i=1 2 (7.4.11b) 2
N32 ( ) = 3( p 1) :
26 (7.4.11c) 22 Analysis of the Finite Element Method...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
- Spring '14