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Mathematics of Computation

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

Authors: Peter K. Moore and Marina Rangelova
Journal: Math. Comp. 79 (2010), 677-705
MSC (2000): Primary 65M60; Secondary 65M15
Published electronically: July 22, 2009
MathSciNet review: 2600539
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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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Additional Information

Peter K. Moore
Affiliation: Department of Mathematics, Southern Methodist University, Dallas, Texas 75275

Marina Rangelova
Affiliation: eVerge Group, Plano, Texas 75093

Keywords: A posteriori error estimation, fourth-order equations, adaptivity
Received by editor(s): December 11, 2007
Received by editor(s) in revised form: April 14, 2009
Published electronically: July 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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