# We also list an estimate of the number of unknowns

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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 higher-order e ect when n is large. The asymptotic approximation N (pn)2 is recorded in Table 5.5.1. Similarly, bi-polynomial 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 bi-polynomial 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 bi-polynomial 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 Stencil 11 00 11 00 11 00 11 00 11 00 11 00 23 11 00 11 00 11 00 1111 0000 1111 0000 1111 0000 11 00 11 00 11 00 11 11 00 00 11 11 00 00 11 11 00 00 11 00 11 00 11 00 11 00 11 00 11 00 11 11 00 00 11 11 00 00 11 11 00 00 11 00 11 00 11 00 11 1111 11 11 00 0000 00 00 11 1111 11 11 00 0000 00 00 11 1111 11 11 00 0000 00 00 3 11 00 11 00 11 00 11 00 11 00 11 00 1111 0000 1111 0000 1111 0000 11 11 00 00 11 11 00 00 11 1111 00 0000 11 00 11 11 00 00 11 1111 00 0000 11 00 11 1111 00 0000 11 00 11 1111 1111 11 11 00 0000 0000 00 00 11 1111 1111 11 11 00 0000 0000 00 00 11 1111 1111 11 11 00 0000 0000 00 00 4 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 11 00 00 11 00 11 00 11 11 00 00 11 11 00 00 11 00 11 11 00 00 11 11 00 00 11 00 11 11 00 00 11 11 00 00 11 11 11 00 00 00 11 11 00 00 11 00 11 11 00 00 11 11 00 00 11 00 11 11 11 00 00 00 11 11 11 11 11 11 00 00 00 00 00 00 11 11 1111 00 00 0000 11 11 1111 11 11 00 00 0000 00 00 11 11 00 00 11 11 00 00 11 11 00 00...
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