A posteriori error estimation for ℎ𝑝-adaptivity for fourth-order equations

Moore, Peter; Rangelova, Marina

doi:10.1090/S0025-5718-09-02290-X

A posteriori error estimation for $hp$-adaptivity for fourth-order equations
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by Peter K. Moore and Marina Rangelova PDF

Math. Comp. 79 (2010), 677-705 Request permission

Abstract:

A posteriori error estimates developed to drive hp-adaptivity for second-order reaction-diffusion equations are extended to fourth-order equations. A $C^1$ hierarchical finite element basis is constructed from Hermite-Lobatto polynomials. A priori estimates of the error in several norms for both the interpolant and finite element solution are derived. In the latter case this requires a generalization of the well-known Aubin-Nitsche technique to time-dependent fourth-order equations. We show that the finite element solution and corresponding Hermite-Lobatto interpolant are asymptotically equivalent. A posteriori error estimators based on this equivalence for solutions at two orders are presented. Both are shown to be asymptotically exact on grids of uniform order. These estimators can be used to control various adaptive strategies. Computational results for linear steady-state and time-dependent equations corroborate the theory and demonstrate the effectiveness of the estimators in adaptive settings.

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