In order to specify this more precisely let nbj k k 1

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Unformatted text preview: let nbj k , k = 1 2 : : : NE , denote the indices of the NE elements sharing the bounding faces of j and let @ j k , k = 1 2 : : : NE , be the faces of j (Figure 10.3.1). Then, we write (10.3.2a) in the more explicit form N X (v ut)j + < v F n(Uj Unb ) >j k ;(rv f )j = (v b)j 8v 2 L2( j ): (10.3.3) E k=1 jk 34 Hyperbolic Problems Ωnb Ωnb j,3 j,2 Ωj Ωnb j,1 Figure 10.3.1: Element j and its neighboring elements indicating that the segments @ k = 1 2 : : : NE , . j k, Without the need to maintain inter-element continuity, virtually any polynomial basis can be used for the approximate solution Uj (x y z t) on j . Tensor products of Legendre polynomials can provide a basis on square or cubic canonical elements, but these are unavailable for triangles and tetrahedra. Approximations on triangles and tetrahedra can use a basis of monomial terms. Focusing on two-dimensional problems on the canonical (right 45 ) triangle, we write the nite element solution in the usual form Uj (x y t) = n X p k=1 ckj Nk ( ) (10.3.4) where np = (p + 1)(p + 2)=2 is the number of monomial terms in a complete polynomial of degree p. A basis of monomial terms would set N1 = 1 N2 = N3 = ::: Nn = p: p (10.3.5) 10.3. Multidimensional Discontinuous Galerkin Methods 35 All terms in the mass matrix can be evaluated by exact integration on the canonical triangle (cf. Problem 1 at the end of this section) as long as it has straight sides however, without orthogonality, the mass matrix will not be diagonal. This is not a severe restriction since the mass matrix is independent of time and, thus, need only be inverted (factored) once. The ill-conditioning of the mass matrix at high p is a more important concern with the monomial basis (10.3.5). Ill-conditioning can be reduced and the mass matrix diagonalized by extracting an orthogonal basis from the monomial basis (10.3.5). This can be done by the GramSchmidt orthogonalization process shown in Figure 10.3.2. The inner product and norm are de ned in L2 on...
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