Abstract: A new fast multipole boundary element method is proposed for analyzing 2-D orthotropic potential problems by using linear and three-node quadratic elements. In the fast multipole boundary element method, fast multipole expansions are used for the integrals on elements that are far away from the source point, and the direct evaluations are used for the integrals on elements that are close to the source point. The use of linear and three-node quadratic elements results in more complicated computations for near-field integrals, especially singular integrals and nearly singular integrals. In this paper, the complex notation is introduced to simplify the near-field integrals. If the boundary is discretized by linear elements, the near-field integrals are calculated by the analytic formulas, if the three-node quadratic element is used, a semi analytical algorithm is given to calculate the near-field integrals. Accurate evaluations of the singular integrals and nearly singular integrals on linear and three-node quadratic elements ensure that the present fast multipole boundary element method can be applied to the ultra-thin structure, which broadens the application of the fast multipole boundary element method with linear and quadratic elements. Numerical examples show that the number of elements required by the fast multipole boundary element method with linear and quadratic elements is significantly less than that with constant elements. In addition, the required CPU time is increased linearly with the increase of the number of degrees of freedom (N), which demonstrates the computational efficiency is still in the complexity of O (N). Therefore, the present method exhibits higher accuracy and efficiency for solving large-scale problems.