Analysis of singularity in advection-diffusion-reaction equation with semi-analytical boundary elements

In order to solve the advection-diffusion-reaction equation with singularity by the boundary element method, the domain integral is handled by dual reciprocity method and the semi-analytical elements are implemented to approximate the unknown physical fields. By collocating internal points in the domain and approximating the primer field with radial basis function, the integral of domain can be settled and boundary-only integral equation can be derived. In this paper, we propose two new elements containing asymptotic expressions to model the unknown temperature and singular flux fields, while the known fields are simulated by linear interpolation. The unknowns in the new elements are amplitude coefficients which can be used to evaluate the intensity of flux. Compared with conventional boundary element, the proposed method allows to avoid direct evaluation of singular values. Thus, the accuracy can be guaranteed since all the unknowns have finite values. Benefiting from the semi-analytical elements, the mesh refinement for the region with singular physical field can be avoided. Moreover, the first three amplitude coefficients as well as the whole physical fields are obtained at the same time without any post process. We demonstrate the excellent numerical results through several examples in plates with singularities.