Error Analysis on Galerkin Scheme for the Diffusion Problem

For advection-diffusion problem, the non-unified approach of classic residual distribution (RD) schemes for advection discretization and pure Galerkin scheme for diffusion discretization has seen accuracy degradation and previous work has dealt with the accuracy analysis of the advection part on various RD schemes. The current work focuses on the accuracy analysis on diffusion discretization scheme of pure Galerkin method with the effect of grid skewness variation. Analytically, the accuracy of the pure Galerkin method is analyzed rigorously with Fourier expansion technique for pure diffusion problem in right-running grid with the introduction of grid skewness parameter as a variable. The truncation error study shows that pure Galerkin scheme is second order accurate in space with minimal or no effect from the variation of skewness. Numerical tests were performed on both steady and unsteady diffusion problems and the outcome matches the analytical study. Additional numerical test on randomized grid revealed that the analytical claims on uniform grid can be extended to poor quality grid. This work suggests that the diffusion discretization using pure Galerkin scheme is not the main factor of the degradation of the accuracy of the non-unified RD-Galerkin scheme.