A posteriori pointwise error estimates for the boundary element method

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].…”

“…According to the definition, e(x) in [8] (18), it can be found that E t (x) and E u (x) as defined by Paulino et al [8,9] are, respectively, {t UT x Ϫ t LM x} and {u UT x Ϫ u LM x}; if the discretization procedure is employed. …”

“…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.…”