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 interelement 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 twodimensional 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 illconditioning of the mass matrix at high p is a more
important concern with the monomial basis (10.3.5).
Illconditioning 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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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, Constant of integration, Partial differential equation, hyperbolic problems

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