Investigation of diffusion in p-version ‘LSFE’ and ‘STLSFE’ formulations

This paper deals with investigation of diffusion for p-version least squares finite element formulation (LSFEF) and p-version space-time coupled least squares finite element formulation (STLSFEF) for steady-state and transient problems. Convection dominated flows result in hyperbolic system of equations which leads to ill-conditioned matrices when using Galerkin formulation. Various techniques (SUPG, SUPG-with discontinuity capturing operator etc.) have been devised to overcome the difficulties arising primarily due to hyperbolic terms and sharp gradients. In this paper, it is demonstrated that when using p-version STLSFEF or LSFEF, no such difficulties are encountered in formulation as well as in the solution procedure. Almost all numerical processes suffer from numerical diffusion to some extent, however, it is demonstrated in this paper that in p-version STLSFE and LSFE formulations numerical diffusion can be completely eliminated by mesh refinement and p-level increase and the formulations are free of inherent diffusion. Several model problems are considered with dominant convective terms to investigate diffusion in p-version LSFEF and STLSFEF. Two dimensional convection-diffusion problems are used as steady state representative cases. One dimensional transient problems considered in this paper include pure advection, convection-diffusion and Burgers' equation. Numerical results are also compared with exact solutions and those reported in the literature.