INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (in press) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1613 Galerkin boundary integral analysis for the axisymmetric Laplace equation ‖ L. J. Gray 1, 3, ∗, † , Maria Garzon 2, ‡ , Vladislav Mantiˇ c 3, § and Enrique Graciani 3, ¶ 1 Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A. 2 Department of Applied Mathematics, University of Oviedo, Oviedo, Spain 3 Elasticity and Strength of Materials Group, University of Seville, Camino de los Descubrimientos, s/n, 41092 Seville, Spain SUMMARY The boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green’s function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric-Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non-singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post-processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd. KEY WORDS: axisymmetric Laplace; boundary integral equation; Galerkin approximation; singular integration ∗ Correspondence to: L. J. Gray, Computer Science and Mathematics Division, ORNL, Oak Ridge, TN 37831, U.S.A. † E-mail: ljg@ornl.gov, graylj1@ornl.gov ‡ E-mail: maria@orion.ciencias.uniovi.es § E-mail: mantic@esi.us.es ¶ E-mail: graciani@esi.us.es ‖ This article is a U.S. Government work and is in the public domain in the U.S.A. Contract/grant sponsor: Spanish Ministry of Education, Culture and Sport; contract/grant number: SAB 2003- 0088 Contract/grant sponsor: Spanish Ministry of Science and Technology; contract/grant numbers: MAT 2003-03315, MEC-04-MTM-05417 Contract/grant sponsor: U.S. Department of Energy; contract/grant number: DE-AC05-00OR22725 Received 15 July 2005 Revised 31 October 2005 Published in 2005 by John Wiley & Sons, Ltd. Accepted 9 November 2005