Hypersingular Residuals—a New Approach for Error Estimation in the Boundary Element Method

Abstract

SUMMARYThis paper presents a new approach for a posteriori 'pointwise' error estimation in the boundary element method. The estimator relies upon evaluation of the residual of hypersingular integral equations, and is therefore intrinsic to the boundary integral equation approach. A methodology is developed for approximating the error on the boundary as well as in the interior of the domain. Extensive computational experiments have been performed for the two-dimensional Laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve. Moreover, a reasonable estimate of the magnitude of the actual error is also predicted.

Error estimation for boundary element method

“…We investigate this new error estimator ͉u UT x Ϫ u LM x͉ or ͉t UT x Ϫ t LM x͉ for comparison with both the error estimator e(x) and exact error E(x) defined by Paulino et al [8,9].…”

“…The discretization error is defined as the difference between the exact solution and the numerical approximation of the governing equation. Obtaining a reliable error estimation [1][2][3][4][5][6][7][8][9][10][11][12] is very important in order to guarantee a certain level of accuracy of the numerical result, and is a key factor of the adaptive mesh procedure [3][4][5][6][13][14][15][16][17]. Thus, estimation of the discretization error in the Boundary Element Method (BEM) is worthy of study.…”

“…However, the error stems not only from the discretization procedure, but also from the mismatch of the collocation points on the boundary. Zarikian et al [7] and Paulino et al [8,9] used pointwise error estimation to study the convergency of the interior problem by using the dual BEM. Both the first (singular integral equation) and second (hypersingular integral equation) kind of formulation of BEM can independently determine the unknown boundary data for the problem without degenerate boundary [19].…”

“…Two numerical examples are performed for twodimensional problems of the Laplace equation with the Dirichlet and mixed boundary conditions using the BEPO2D program [19]. For an interior problem, Paulino et al [8,9] employed the direct BEM to solve the unknown boundary data, u UT , in the Neumann problem and t UT in the Dirichlet problem, and then substituted u UT or t UT into the hypersingular integral equation. Inequality occurs in Eq.…”

Adaptive meshing for two-dimensional thermoelastic problems using Hermite boundary elements

“…[24], Paulino et al proposed the evaluation of the residual of HBIEs as an error indicator. The method presented by them was developed for the approximation of the error on the boundary as well as in the interior of the domain.…”