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Unformatted text preview: lds no improvement in the convergence rate.
Example 7.3.2. Problems with variable Jacobians are more complicated. Consider
the term
det(J( ))Wx( ) = J (W x + W x)
where J = det(J( )). The metrics x and x are obtained from the inverse Jacobian
1y
J;1 = x y = J ;y ;x :
x
xy
In particular, x = y =J and x = ;y =J and
In particular, convergence does not occur if det(J)Wx = W y ; W y :
Consider an isoparametric transformation of degree p. Such triangles or quadrilaterals
in the computational plane have curved sides of piecewise polynomials of degree p in the
physical plane. If W is a polynomial of degree p then Wx has degree p ; 1. Likewise,
x and y are polynomials of degree p in and . Thus, y and y also have degrees
p ; 1. Therefore, JWx and, similarly, JWy have degrees r1 = 2(p ; 1). With J being a
polynomial of degree 2(p ; 1), we nd JW to be of degree r0 = 3p ; 2.
For the quadrature errors (7.3.6) to have the same O(hp) rate as the interpolation
error, we must have r = p ; 1 in (7.3.6a,b). Thus, according to Theorem 7.3.2, the order
of the quadrature rules in the (...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

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