Negative Norm Estimates and Superconvergence
In this chapter we shall extend our earlier error estimates in L2 and H1 to estimates in norms of negative order. It will turn out that if the accuracy in L2 of the family of approximating spaces is O(h r) with r > 2, then the error bounds in norms of negative order is of higher order than O(h r). In certain situations these higher order bounds may be applied to show error estimates for various quantities of these higher orders, so called superconvergent order estimates. We shall exemplify this by showing how certain integrals of the solution of the parabolic problem, and, in one space dimension, the values of the solution at certain points may be calculated with high accuracy using the semidiscrete solution.
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