A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity

In this paper a boundary-domain integral diffusion–convection equation has been developed for problems of spatially variable velocity field and spatially variable coefficient. The developed equation does not require a calculation of the gradient of the unknown field function, which gives it an advantage over the other known approaches, where the gradient of the unknown field function is needed and needs to be calculated by means of numerical differentiation. The proposed equation has been discretized by two approaches—a standard boundary element method, which features fully populated system matrix and matrices of integrals and a domain decomposition approach, which yields sparse matrices. Both approaches have been tested on several numerical examples, proving the validity of the proposed integral equation and showing good grid convergence properties. Comparison of both approaches shows similar solution accuracy. Due to nature of sparse matrices, CPU time and storage requirements of the domain decomposition are smaller than those of the standard BEM approach.