A new approach is presented, in which thenonsingular IBIE is established. A numerically systematic scheme isestablished by adopting quadratic Lagrange's elements.A new geometry boundary approximatetechnique is presented. Some numerical examples will be applied to validate the current scheme. It is shown that a betterprecision and high computational efficiency can be derived by thepresentation, especially the numerical results of boundary quantities match theexact solutions very well. Besides, it can be easily extended to the threedimensional problems.Compared with the direct unknown's situation, the present approach has manyadvantages:1) it is not necessary to deal with the HFP so that it is simpler,and with better precision and high computational efficiency. 2) The presentation only requires algebraic manipulations so that it can easilybe extended to the other problems, such as potential problems and Stokesproblems. 3) This paper establishes a technique applicable for thecalculation of CPV integrals.