Pii: S0955-7997(98)00086-1

In this paper, six error indicators obtained from dual boundary integral equations are used for local estimation, which is an essential ingredient for all adaptive mesh schemes in BEM. Computational experiments are carried out for the two-dimensional Laplace equation. The curves of all these six error estimators are in good agreement with the shape of the error curve. The results show that the adaptive mesh based on any one of these six error indicators converges faster than does equal mesh discretization. q 1999 Elsevier Science Ltd. All rights reserved. Keywords: Dual boundary integral; Error estimator; Adaptive technique 1. Introduction The numerical method can be utilized to solve the governing equation of a problem especially when the exact solution is not easy to obtain. However, the discretization process, which transforms a continuous system into a discrete system with finite number of degrees of freedom, results in errors. 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 [1–12] 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 [3–6,13– 17]. Thus, estimation of the discretization error in the Boundary Element Method (BEM) is worthy of study. The h-refinement [5,6], p-refinement [5,18] and r-refinement schemes [5] have been recently used to improve numerical accuracy. In the h-refinement scheme, the total number of elements increase, but the order of the interpolation function remains unchanged. As the global matrix must be reformulated after mesh refinement, the computational cost becomes very high. In this way, the efficient remesh tactics are required when h-refinement scheme is adopted. The adaptive tactics for h-refinement are generally referred to as the reference value method [5], in which the element mesh is refined where the error is larger than the prescribed reference value. This method provides a facile error criterion to determine which elements should be divided into more partitions by considering the integral equation at the sampling point. A large number of studies on adaptive BEM have been done by Kamiya et al. [4] using sample point error estimation. 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]. The difference in the solutions obtained from these two methods can be used as an index of error estimation. This provides a new guide for remeshing without the mismatch of the collocation points on the boundary in the sample point error method. By creating more divisions in the boundary mesh where the estimated error is large, the exact error will be reduced more efficiently. In this paper, dual boundary integral equations are utilized to find a reliable error estimator. The estimated error is defined as the residue [10]. The residue is based on the imbalance of the energy, which is calculated by substituting the solution of the unknown boundary data obtained from the first kind of direct BEM previously and known boundary data into the hypersingular integral representation. Two numerical examples are performed for twodimensional problems of the Laplace equation with the Dirichlet and mixed boundary conditions using the BEPO2D program [19]. Engineering Analysis with Boundary Elements 23 (1999) 257–265 0955-7997/99/$ see front matter q 1999 Elsevier Science Ltd. All rights reserved. PII: S0955-7997(98)00086-1 * Corresponding author. Fax: 1 886-2-2462-2192-6102. 2. Review of a posteriori pointwise error estimator for the boundary element method Considering a two-dimensional Laplace equation, as the one below, 7u…x† ˆ 0; x [ D …1† where D is the domain, and u(x) is the potential function, the first kind of BEM (direct BEM) for this problem can be written as