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Unformatted text preview: 11 11 00 00 np N p2 n2 3 n2 6 4n2 10 9n2 15 16n2 Table 5.5.1: Shape function placement for Lagrange and hierarchical nite element approximations of degrees p = 1 2 3 4 on triangular elements with their number of parameters per element np and degrees of freedom N on a square with 2n2 elements. Circles indicate additional shape functions located on a mesh entity. of accuracy is a complex issue that depends on solution smoothness, geometry, and the partial di erential system. We'll examine this topic in a later chapter. At least it seems clear that bi-polynomial bases are not competitive with hierarchical ones on square elements. 5.5.1 Generation of Element Matrices and Vectors The generation of the element sti ness and mass matrices and load vectors is largely independent of the partial di erential system being solved however, let us focus on the model problem of Section 3.1 in order to illustrate the procedures less abstractly. Thus, 1 consider the two-dimensional Galerkin problem: determine u 2 HE satisfying A(v u) = (v f ) 1 8v 2 H0 (5.5.1a) 24 Mesh Generation and Assembly p 100 11 11 00 11 00 11 00 11 200 11 00 11 00 11 00 11 00 11 00 11 00 11 00 Lagrange Hierarchical 2 Stencil np N M n Stencil np N M n2 n2 00 n2 1 04 11 1 04 1 0 11 00 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11 00 11 00 111 1 000 0 111 1 000 0 11 111 1 00 000 0 11 111 1 00 000 0 11 111 1 00 000 0 11 111 1 00 000 0 11 111 1 00 000 0 11 111 1 00 000 0 11 111 1 00 000 0 11 011 1 1 400 0 00 0 11 1 1 00 0 0 11 00 11 1 1 00 1 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 00 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 1 1 1 00 0 0 0 11 00 11 300 11 00 11 00 11 00 9 4n 16 9n2 2 1 0 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 11 00 11 00 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 11 0 00 1 11 0 00 1 0 1 0 16n2 25 1 11 0 00 1 11 0 00 1 11 0 00 3n2 12 5n2 17 8n2 11 00 11 00 1 0 1 0 1 0 8 11 00 11 00 11 00 1 0 1 0 1 0 1 0 1 0 1 0 11 00 11 00 1 0 1 0 1 0 1 0 11 00 11 00 1 0 1 0 1 0 1 0 Table 5.5.2: Shape function placement for bi-polynomial Lagrange and hierarchical approximations of degrees p = 1 2 3 4 on square elements with their number of parameters per element np and degrees of freedom N on a square with n2 elements. Circles indicate additional shape functions...
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