Invited Paper A two-step grid optimization method for the direct boundary element method

Most grid optimization methods for the boundary element method are based on a postieri error indicators and error estimators. The error indicators are a local estimate of error within each boundary element. The error estimator provides a global bound on the error and is calculated from the error indicators. Typically, a problem is run on a coarse grid and the local error indicators are calculated within each element. Based on the error indicators, a threshold value for the error indicators is selected. In an adaptive grid optimization method, those elements whose error indicators are above the threshold value are either subdivided in the h-method approach or enriched by increasing the order of the polynomial approximation within the element in the p-method approach. The problem is then resolved on the finer mesh. One problem with these approaches is choosing the threshold error tolerance. If this value is too high, then very few degrees of freedom are added at each stage of the mesh refinement. If the value is too low, then the method becomes less adaptive. We discuss in this paper a two-step grid optimization approach which does not require the selection of a threshold error tolerance. In this method, the problem is run on a coarse mesh to determine the error indicators. Based solely on these error indicators, the remaining degrees of freedom are incorporated into the discretization. This two-step grid optimization procedure is compared to the more traditional approaches based on selecting a threshold error tolerance.