4 interpolation error estimates interpolation error

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Unformatted text preview: 0] describe many of their properties, similarities, and di erences. Let us set the stage by brie y describing two simple extrapolation techniques. Consider a p one-dimensional problem for simplicity and suppose that an approximate solution Uh (x) has been computed using a polynomial approximation of degree p on a mesh of spacing h (Figure 7.4.1). Suppose that we have an a priori interpolation error estimate of the form p u(x) ; Uh (x) = Cp+1hp+1 + O(hp+2): We have assumed that the exact solution u(x) is smooth enough for the error to be expanded in h to O(hp+2). The leading error constant Cp+1 generally depends on (unknown) derivatives of u. Now, compute a second solution with spacing h=2 (Figure 7.4.1) to obtain p u(x) ; Uh=2 (x) = Cp+1( h )p+1 + O(hp+2): 2 18 Analysis of the Finite Element Method U2 h U1 h/2 U1 h h x 1 1 Figure 7.4.1: Solutions Uh and Uh=2 computed on meshes having spacing h and h=2 with 2 piecewise linear polynomials (p = 1) and a third solution Uh computed on a mesh of spacing h with a piecewise quadratic polynomial (p = 2). Subtracti...
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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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