Then 1 if a quadrature rule is exact in the

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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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