Galerkin weighted residual method with high-order trial functions for convection diffusion problem

Galerkin weighted residual method with high-order trial functions for convection diffusion problem

A stabilized Galerkin weighted residual method using high-order trial functions is proposed to obtain closed-form solutions for one-dimensional convection-diffusion equation at high Peclet numbers. In this method, an approximate solution, written as a linear expansion of known global trail functions multiplied by unknown coefficients, is substituted into the differential equation yielding a residual. Subsequently, the integral over the problem domain of the residual weighted by each of the trial functions is set to zero resulting in a system of algebraic equations for the unknown coefficients. Solving the system of equations analytically yields a closed-form solution for the problem. To obtain stable solutions for convection-domained cases, high order polynomials are employed as trial functions. The obtained results are in close agreements with the analytical solution for a wide range of Peclet numbers