A critical study of hypersingular and strongly singular boundary integral representations of potential gradient

Abstract

This paper deals with recovery of potential gradient (∇u) in the resolution of the two-dimensional Laplace equation by boundary element method. The emphasis is placed on ∇u-recovery near and on the boundary, where error behaviour and convergence rate are studied in detail. Conventional hypersingular (HS) and strongly singular (SS) boundary integral representations (BIRs) of ∇u are compared with two recovery procedures recently introduced by Mantič, Graciani and París, Comput. Methods Appl. Mech. Engrg (1999) 178, pp. 267–289: DSC – a local smoothing procedure, and SSC - applying SS BIR with the integral density obtained previously from DSC. The theoretical analysis of error of ∇u recovered by ∇u-BIRs presented is supported by numerical examples. The numerical study of convergence carried out using an h-refinement of (quasi)uniform meshes of linear boundary elements shows a slow convergence rate, O(h) or less, of HS and SS BIRs in comparison with superconvergence rate O(h ²) of DSC (only on the boundary) and SSC near the boundary and at element centres and junctions between elements. A slow convergence rate is obtained by all approximations at boundary corners, SSC clearly providing the most accurate results. Superconvergent results of HS and SS BIRs are only obtained in a particular case of evaluation of tangential derivative of potential on a straight part of the boundary.