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41 solutions uh and uh2 computed on meshes having

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Unformatted text preview: ng the two solutions we eliminate the unknown exact solution and obtain p p Uh=2 (x) ; Uh (x) = Cp+1hp+1(1 ; 2p 1 1 ) + O(hp+2): + Neglecting the higher-order terms, we obtain an approximation of the discretization error as p p p+1 Uh=2 (x) ; Uh (x) : Cp+1h 1 ; 1=2p+1 Thus, we have an estimate of the discretization error of the coarse-mesh solution as p p p (x) Uh=2 (x) ; Uh (x) : u(x) ; Uh 1 ; 1=2p+1 The technique is called Richardson's extrapolation or h-extrapolation. It can also be used to obtain error estimates of the ne-mesh solution. The cost of obtaining the error estimate is approximately twice the cost of obtaining the solution. In two and three dimensions the cost factors rise to, respectively, four and eight times the solution cost. Most would consider this to be excessive. The only way of justifying the procedure is to consider the ne-mesh solution as being the result and the coarse-mesh solution as furnishing the error estimate. This strategy only furnishes an error estimate on the coarse mesh. A...
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