Unformatted text preview: for all polynomials of degree at most r0 + r ; 1,
j(f W ) ; (f W ) j
N
C hr+1 kf kr+1
8W 2 S0 :
(7.3.6b)
kW k1
Proof. cf. Wait and Mitchell 21], Chapter 6, or Strang and Fix 18], Chapter 4.
Example 7.3.1. Suppose that the coordinate transformation is linear so that det(J( ))
N
is constant and that S0 consists of piecewise polynomials of degree at most p. In this
case, r1 = p ; 1 and r0 = p. The interpolation error in H 1 is
ku ; V k1 = O(hp): Suppose that the quadrature rule is exact for polynomials of degree or less. Thus,
= r1 + r or r = ; p + 1 and (7.3.6a) implies that
jA(V W ) ; A (V W )j
N
C h ;p+2kV k ;p+3
8V W 2 S0 :
kW k1
With = r0 + r ; 1 and r0 = p, we again nd r = ; p + 1 and, using (7.3.6b),
j(f W ) ; (f W ) j
N
C h ;p+2kf k ;p+2
8 W 2 S0 :
kW k1
If = 2(p;1) so that r = p;1 then the above perturbation errors are O(hp). Hence,
all terms in (7.3.3) or (7.3.5) have the same order of accuracy and we conclude that
ku ; U k1 = O(hp): This situation is regarded as optimal. If the coe cients of the di erential equation
are constant an...
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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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