Residual-free bubble shape functions used in BEM for the stability of the solution of the convective-diffusion transport equation

Abstract

In this paper we focus on oscillation and instability of the solution of the convective-diffusive equation depending on the Peclet number. For that purpose, different types of mesh elements and shape functions have been used. For the stabilization of the solution of the convective-diffusive equation for high Peclet numbers, we employed high-order polynomial shape functions, namely residualfree bubble functions. The numerical scheme is based on BEM. For solving the linear homogeneous part of the partial differential equation, the Laplace fundamental solution has been used.We compared quadratic nine-node domain element using Lagrangian shape functions, linear four-node domain element using Lagrangian shape functions and linear four-node domain element using fourth-order bubble enriched functions. Numerical results obtained with linear Lagrangian shape function and bubble enriched functions are compared with the analytical solution.Residual-free bubble functions add stability to simulation and despite the fact that less nodes are used in the domain element, the results are comparable and in some cases even better than the quadratic nine-node domain element. The boundary element method with usage of bubble-enriched functions can resolve problems of convective-diffusion and obtain stable and accurate solutions for this type of governing equations, which are being represented in several types of transport phenomena.