Estimating the optimum number of boundary elements by error estimation in a defined auxiliary problem

Abstract

In this paper, we develop a new estimation technique to estimate the numerical error of the boundary element method (BEM). The discretization error of the BEM can be evaluated without having analytical solution. An auxiliary problem is defined to substitute for the original problem. In the auxiliary problem, the governing equation (GE), domain shape and boundary condition (BC) type are the same as the original problem. By using the linear combination of the complementary solutions set of GE, the analytical solution of the auxiliary problem, which is named quasi-analytical solution is defined. The quasi-analytical solution satisfies the GE and is similar to the real analytical solution of the original problem. By implementing the BEM to solve the auxiliary problem and comparing with the analytical solution, the error magnitude is estimated, which is approximate to the real error. The curve of the R.M.S. error versus different number of elements can be obtained. As a result, we can estimate the optimal number of elements in BEM. Several numerical examples are taken to demonstrate the accuracy of the proposed error estimation technique.