Analysis of a consistency recovery method for the 1D convection–diffusion equation using linear finite elements

Abstract

AbstractFor residual‐based stabilization methods such as streamline‐upwind Petrov–Galerkin (SUPG) and finite calculus (FIC), the higher‐order derivatives of the residual that appear in the stabilization term vanish when simplicial elements are used. The sub‐grid scale method using orthogonal sub‐scales (OSS) attempts to recover the lost consistency by using a fine‐scale projected residual in the stabilization term. The FIC method may also be cast into an OSS form with very little manipulation using an auxiliary convective projection equation. This paper discusses the gain/loss by recovering the consistency of the discrete residual in the stabilization terms via the form that includes the convective projection (as in the OSS method). We present the von Neumann analysis of the FIC method with recovered consistency (FIC_RC) for the 1D convection–diffusion problem and we compare it with the standard Bubnov–Galerkin linear finite element method and FIC/SUPG methods. The transient analysis is done by examining the discrete dispersion relation of the stabilization methods. The spectral results for the semi‐discrete and fully discrete problem are presented with time integration done by the trapezoidal and second‐order backward differencing formula schemes. The effect of lumping the effective mass matrix T is considered relative to using a consistent form. The effect of refinement in space and time is also discussed. Finally, an optimal expression for the stabilization parameter for the FIC_RC method on a uniform grid and for the steady state is given and its performance in the transient mode is discussed. Copyright © 2008 John Wiley & Sons, Ltd.