The only way of justifying the procedure is to

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Unformatted text preview: nother strategy for obtaining an error estimate by extrapolation is to compute a second solution using a higher-order method (Figure 7.4.1), e.g., p u(x) ; Uh +1 = Cp+2hp+2 + O(hp+3): Now, use the identity p p p p u(x) ; Uh (x) = u(x) ; Uh +1 (x)] + Uh +1 (x) ; Uh ]: 7.4. A Posteriori Error Estimation 19 The rst term on the right is the O(hp+1) error of the higher-order solution and, hence, can be neglected relative to the second term. Thus, we obtain the approximation p p p u(x) ; Uh (x) Uh +1 (x) ; Uh (x): The di erence between the lower- and higher-order solutions furnish an estimate of the error of the lower-order solution. The technique is called order embedding or p-extrapolation. There is no error estimate for the higher-order solution, but some use it without an error estimate. This strategy, called local extrapolation, can be dangerous near singularities. Unless there are special properties of the scheme that can be exploited, the work involved in obtaining the error estimate is comparable to the work of o...
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