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16 if s n consists of complete piecewise polynomials

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Unformatted text preview: of complete piecewise polynomials of degree p with respect to a sequence of uniform meshes (cf. De nition 4.6.1) and u 2 H p+1. The bound (7.1.6) can be combined with either (7.1.3) or (7.1.5) to provide an estimate of the error and convergence rate of a nite element solution. The Sobolev norm on H 1 and the strain energy norm (7.1.4) are equivalent for the model problem (7.1.1) and we shall use this with (7.1.3) and (7.1.6) to construct error estimates. Prior to continuing, you may want to review Sections 2.6, 3.2, and 4.6. A priori nite element discretization errors, obtained as described, do not account for such \perturbations" as 1. using numerical integration, 2. interpolating Dirichlet boundary conditions by functions in S N , and 3. approximating @ by piecewise-polynomial functions. 7.2. Convergence and Optimality 3 These e ects will have to be appraised. Additionally, the a priori error estimates supply information on convergence rates but are di cult to use for quantitative error information. A posteriori error...
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