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IMA Journal of Numerical Analysis Advance Access published online on September 26, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp023
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

An a posteriori error indicator for discontinuous Galerkin approximations of fourth-order elliptic problems

Emmanuil H. Georgoulis{dagger}

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

Paul Houston{ddagger}

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK

Juha Virtanen§

Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, UK

{dagger} Corresponding author. Email: emmanuil.georgoulis{at}mcs.le.ac.uk

{ddagger} Email: paul.houston{at}nottingham.ac.uk

§ Email: jmv8{at}le.ac.uk

Received on 5 June 2008. Revised on 19 May 2009.


  Abstract

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems.

Key Words: Discontinuous Galerkin methods; biharmonic problem; fourth order PDEs; a posteriori error analysis; adaptivity


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