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Unformatted text preview: umber of degrees of freedom is N = (pn + 1)2 (cf. Problem 1 at the
end of this section). Dirichlet data on the entire boundary would reduce N by O(pn)
and, hence, be a higherorder e ect when n is large. The asymptotic approximation
N (pn)2 is recorded in Table 5.5.1. Similarly, bipolynomial approximations of order p
on squares with n2 uniform elements have N = (pn + 1)2 degrees of freedom (again, cf.
Problem 1). The asymptotic approximation (pn)2 is reported in Table 5.5.2. Under the
same conditions, hierarchical bases on squares have
; 1)n2 + pn + 1
N = (2p ; p + 4)n2=2 + 2pn + 1 iif p < 4 :
2
2
(p
fp 4
degrees of freedom. The asymptotic values N (2p ; 1)N 2 , p < 4, and N (p2 ; p +
4)n2=2, p 4, are reported in Table 5.5.2.
The Lagrange and hierarchical bases on triangles and the Lagrange bipolynomial
bases on squares have approximately the same number of degrees of freedom for a given
order p. The hierarchical bases on squares have about half the degrees of freedom of the
others. The bipolynomial Lagrange shape functions on a square have the largest number
of parameters per element for a given p. The number of parameters per element a ects the
element matrix and vector computations while the number of degrees of freedom a ects
the solution time. We cannot, however, draw rm conclusions about the superiority of
one basis relative to another. The selection of an optimal basis for an intended level 5.5. Element Matrices and Their Assembly p Lagrange
Stencil 1 2 Hierarchical
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 Spring '14
 JosephE.Flaherty

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