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49a 2 where rx y f r pru qu 749b is the residual

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Unformatted text preview: ere are no errors in the solution subspace. Bank 10] chose S N as a space of N 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 N 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 + : 2 (7.4.10a) (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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