# Then 1 if a quadrature rule is exact in the

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

Ask a homework question - tutors are online