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IMA Journal of Numerical Analysis Advance Access published online on October 6, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp028
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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.

Pointwise error estimate and asymptotic error expansion inequalities for a stabilized Galerkin method

Jaeun Ku{dagger}

Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences Building, Stillwater, OK 74078-1058, USA

{dagger} Email: jku{at}math.okstate.edu

Received on 28 April 2008. Accepted for publication 8 July 2009.


  Abstract

This paper contains new pointwise error estimates for a stabilized Galerkin method proposed by Bramble et al. (1998, Comput. Methods Appl. Mech. Eng., 152, 195–210) and Ku (2007, Math. Comput., 76, 97–114) for second-order elliptic partial differential equations. The estimates show a local dependence of the error at a point on the derivative of the solution u and a weak dependence on the global norm. The results in this paper are stronger than the maximum norm error estimates in Ku (2007). As elementary consequences of the new pointwise error estimates, we provide asymptotic error expansion inequalities by following the idea of Schatz (1998, Math. Comput., 67, 877–899). The results are valid for large classes of finite-element spaces on irregular grids.

Key Words: stabilized Galerkin method; error estimates; error expansion inequality


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