On streamline diffusion arising in Galerkin FEM with predictor/multi‐corrector time integration

Abstract

SUMMARYIn the present paper, the author shows that the predictor=multi-corrector (PMC) time integration for the advection-di usion equations induces numerical di usivity acting only in the streamline direction, even though the equations are spatially discretized by the conventional Galerkin ÿnite element method (GFEM). The transient 2-D and 3-D advection problems are solved with the PMC scheme using both the GFEM and the streamline upwind=Petrov Galerkin (SUPG) as the spatial discretization methods for comparison. The solutions of the SUPG-PMC turned out to be overly di usive due to the additional PMC streamline di usion, while the solutions of the GFEM-PMC were comparatively accurate without signiÿcant damping and phase error. A similar tendency was seen also in the quasi-steady solutions to the incompressible viscous ow problems: 2-D driven cavity ow and natural convection in a square cavity.

A new positive‐definite regularization of incompressible Navier–Stokes equations discretized with Q1/P0 finite element

“…To escape from the Babuska-Brezzi test and to manage to achieve the complete uniqueness of the Q1=P0 solutions in another way, the present author newly proposes the following regularization for Equation (8).…”

“…The lumped mass matrix is thoroughly used to avoid the full inversion of the consistent mass matrix, except for the mass matrix multiplication appearing in Step 1 in the multi-corrector phase as seen in Table I. The participation of the consistent mass matrix helps to reduce numerical errors inherent in fully explicit methods with the mass lumping [7,8].…”

Variational multiscale stabilized FEM formulations for transport equations: stochastic advection–diffusion and incompressible stochastic Navier–Stokes equations

“…The transient advection of a cosine hill in a rotating flow field has been identified as a standard test problem for testing advection-diffusion algorithms [48]. The problem definition is depicted in Fig.…”