A note on upwinding and anisotropic balancing dissipation in finite element approximations to convective diffusion problems

Abstract

SUMMARYIn one dimension, Petrov-Galerkin nonsymmetric weighting for the convective diffusion equation can be interpreted as an added dissipation. The addition of an appropriate amount of dissipation can therefore give the same oscillation-free solutions as the 'upwinding', Petrov-Galerkin, finite element methods. The 'balancing dissipation' is optimally chosen so that excessive dissipation does not occur. A scheme is presented for extending this approach to two-dimensional problems, and numerical examples show that the new method can be used with improved computational efficiency.

Development of Computational Techniques for Transonic Flows: An Historical Perspective

“…This method was essentially equivalent to a Galerkin method with linear elements, stabilized by artificial diffusion to produce an upwind bias. Other finite-element methods also have been developed to solve the Euler equations; an example is the the Streamwise Upwind Petrov-Galerkin (SUPG) method (Hughes, Franca, & Mallet [17], Kelly et al [37], Peraire et al [50]; see also Zienkiewicz & Taylor [62]) which automatically incorporates an upwind bias by an appropriate choice of test functions.…”

Thermal Hydraulic Analysis by Skew Upwind Finite Element Method

“…Although some computational methods adopting these measures have been proposed in fact, they are not so suitable for a widely-used computer program. Some comments related to (2) …”

Meshfree Petrov-Galerkin Methods for the Incompressible Navier-Stokes Equations

“…with the help of element matrix and vector norms [24], the Green's function of the element [12], mathematical error analysis [4,5,16], or model equations [2,9,17].…”

“…Then, this solution is equated with the analytical solution of the differential equation [2,9,17]. In the following, we present a new approach which does not require the analytical solution of difference equations.…”

“…Using standard element stabilisation instead of nodal stabilisation with τ e = τ I leads to the same result. This formula for τ has often be called 'optimal' in the literature [1,2,9,17]. It has a local character as it is independent of the boundary conditions and only relies on the relative positions of the neighbouring nodes x I−1 and x I+1 .…”