The Riemann complex boundary element method for the solutions of two-dimensional Elliptic equations

Abstract

In this paper, a new boundary integral equation model Riemann complex boundary element method (RCBEM), is proposed based on the boundary element method (BEM) and the theory of Vekua and its modification as well as complex Riemann function as the fundamental solution. The RCBEM method is used to solve the linear, second order, elliptical partial differential equations in the fluid flow problems. In comparison to the generally used BEM, for RCBEM, there are two distinct differences. First one is that, RCBEM applies complex Riemann function as the fundamental solution of the adjoint operator while in direct BEM, on the other hand Green function is used. The second one is that the governing equations should be transformed into complex domain because there exist two characteristics in complex plane for elliptic systems, while in the direct BEM is not, since the Green function is adopted instead. The singular problem occurring in direct BEM can be avoided in RCBEM, especially for regular domain problems. The efficiency and accuracy of the RCBEM depends on the complex variable integration. To verify the feasibility and accuracy of RCBEM, the model is applied to different case studies of potential flows, Helmholtz equation problem and advection–diffusion problem and results are compared with analytical solutions and other numerical models. The results are satisfactory and prove the applicability of RCBEM for various two-dimensional elliptic equation problems.