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On the sharpness of $ L\sp 2$-error estimates of $ H\sp 1\sb 0$-projections onto subspaces of piecewise, high-order polynomials

Authors: Weimin Han and Søren Jensen
Journal: Math. Comp. 64 (1995), 51-70
MSC: Primary 65N15; Secondary 65N30
MathSciNet review: 1270620
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Abstract: In a plane polygonal domain, consider a Poisson problem $ - \Delta u = f$ with homogeneous Dirichlet boundary condition and the p-version finite element solutions of this. We give various upper and lower bounds for the error measured in $ {L^2}$. In the case of a single element (i.e., a convex domain), we reduce the question of sharpness of these estimates to the behavior of a certain inf-sup constant, which is numerically determined, and a likely sharp estimate is then conjectured. This is confirmed during a series of numerical experiments also for the case of a reentrant corner. For a one-dimensional analogue problem (of rotational symmetry), sharp $ {L^2}$-error estimates are proven directly and via an extension of the classical duality argument. Here, we give sharp $ {L^\infty }$-error estimates in some weighted and unweighted norms also.

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