In the boundary element analysis of the two-dimensional potential problem, the calculation of the boundary potential gradient of the coordinate variable is difficult. Several techniques have been proposed to address this problem so far. However, they require complex theoretical deduction and a large number of numerical manipulation. This paper proposes a new method named auxiliary boundary value problem method (ABVPM) to solve the the gradient boundary integral equation (GBIE) for two-dimensional potential problems. An ABVPM with the same solution domain as the original boundary value problem is constructed, which is an over-determined boundary value problem with known solution. Consequently, the system matrix of the GBIE, which is the most important problem for boundary analysis, will be obtained by solving the ABVPM. It can be used to solve original boundary value problem. The solution procedure is simple, because only a linear system need to be solved to obtain the solution of the original boundary value problem. It is worth noting that it is not necessary to recalculate the system matrix when solving the original boundary value problem, so the efficiency of the auxiliary boundary value method is not very poor. The proposed ABVPM circumvents the troublesome issue of computing the strongly singular integrals, with some advantages, such as simple mathematical deduction, easy programming and high accuracy. More importantly, the ABVPM provides a new way for solving the GBIE. Three benchmark examples are tested to verify the effectiveness of the proposed technique.

In the research of this article, we propose a new boundary-type meshless method---the average source boundary node method for the potential problem(ASBNM). This method is based on the average source technology and couples it with a completely regularized boundary integral equation. In the solution process, this method only uses boundary nodes, without any the concept of unit or integral. In addition, it has the characteristics of simple method and easy programming, so it is suitable for two-dimensional boundary value problems. Numerical examples prove the effectiveness and accuracy of this method.