34a maximizing the inner product ratios over w and

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Unformatted text preview: form );A ku ; U k1 C infN fku ; V k1 + supN jA(V W kW k (V W )j + V 2S0 1 W 2S0 );( sup j(W f kW kW f ) j g: (7.3.5) N 1 W 2S0 To bound (7.3.3) or (7.3.5) in terms of a mesh parameter h, we use standard interpolation error estimates (cf. Sections 2.6 and 4.6) for the rst term and numerical integration error estimates (cf. Chapter 6) for the latter two terms. Estimating quadrature errors is relatively easy and the following typical result includes the e ects of transforming to a canonical element. 7.3. Perturbations 13 Theorem 7.3.2. Let J( ) be the Jacobian of a transformation from a computational N ( )-plane to a physical (x y)-plane and let W 2 S0 . Relative to a uniform family of meshes h, suppose that det(J( ))Wx( ) and det(J( ))Wy ( ) are piecewise polynomials of degree at most r1 and det(J( ))W ( ) is a piecewise polynomial of degree at most r0. Then: 1. If a quadrature rule is exact (in the computational plane) for all polynomials of degree at most r1 + r, jA(V W ) ; A (V W )j N C hr+1kV kr+2 8V W 2 S0 (7.3.6a) kW k1 2. If a quadrature rule is exact...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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