Numerische Mathematik

, Volume 124, Issue 2, pp 309-335

First online:

A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles

  • Carsten CarstensenAffiliated withDepartment of Computational Science and Engineering, Yonsei UniversityInstitut für Mathematik, Humboldt-Universität zu Berlin Email author 
  • , Dietmar GallistlAffiliated withInstitut für Mathematik, Humboldt-Universität zu Berlin
  • , Jun HuAffiliated withLMAM and School of Mathematical Sciences, Peking University Email author 

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The a posteriori error analysis of conforming finite element discretisations of the biharmonic problem for plates is well established, but nonconforming discretisations are more easy to implement in practice. The a posteriori error analysis for the Morley plate element appears very particular because two edge contributions from an integration by parts vanish simultaneously. This crucial property is lacking for popular rectangular nonconforming finite element schemes like the nonconforming rectangular Morley finite element, the incomplete biquadratic finite element, and the Adini finite element. This paper introduces a novel methodology and utilises some conforming discrete space on macro elements to prove reliability and efficiency of an explicit residual-based a posteriori error estimator. An application to the Morley triangular finite element shows the surprising result that all averaging techniques yield reliable error bounds. Numerical experiments confirm the reliability and efficiency for the established a posteriori error control on uniform and graded tensor-product meshes.

Mathematics Subject Classification (2000)

65N10 65N15 35J25