A double-layer interpolation method for implementation of BEM analysis of problems in potential theory

A double-layer interpolation method (DLIM) is proposed to improve the performance of the boundary element method (BEM). In the DLIM, the nodes of an element are sorted into two groups: (i) nodes inside the element, called source nodes, and (ii) nodes on the vertices and edges of the element, called virtual nodes. With only source nodes, the element becomes a conventional discontinuous element. Taking into account both source and virtual nodes, the element becomes a standard continuous element. The physical variables are interpolated by continuous elements (first-layer interpolation), while the boundary integral equations are collocated at the source nodes only. We further established additional constraint equations between source and virtual nodes using a moving least-squares (MLS) approximation (second-layer interpolation). Using these constraints, a square coefficient matrix of the overall system of linear equations was finally achieved. The DLIM keeps the main advantages of MLS, such as significantly alleviating the meshing task, while providing much better accuracy than the traditional BEM. The method has been used successfully for solving potential problems in two dimensions. Several numerical examples in comparison with other methods have demonstrated the accuracy and efficiency of our method.