On spurious oscillations at layers diminishing (SOLD) methods for convection–diffusion equations: Part I – A review

Computable exact bounds for linear outputs from stabilized solutions of the advection–diffusion–reaction equation

“…This stabilization parameter is compared with the well known expression h k /(2|α| k ) (coth((P e) k ) − 1/(P e) k ). In order to compute the previous expression, two different choices for the element size are used: the smallest edge side of the triangle, h 1 k , and the diameter of the element Ω k in the direction of the advection field α, h 2 k , see [24]. These two different expressions to compute the element size yield two different choices of the stabilization parameter, denoted by τ 2 k and τ 3 k respectively.…”

“…Thus, the stabilization parameter is chosen following [22]. However, the error estimation procedure is valid for any choice of the definition of the stabilization parameter, see for instance [23,24,25,26].…”

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

“…Inspired by the bad experiences with the SUPG method and the upwind method, comprehensive numerical studies of stabilized finite element methods for scalar time-dependent convection-diffusion-reaction equations were performed in [17,18]. These studies included besides the SUPG method a number of Spurious Oscillations at Layers Diminishing (SOLD) schemes [11,12], a local projection stabilization (LPS) scheme [25] and two FiniteElement-Method Flux-Corrected-Transport (FEM-FCT) schemes [21,22,23]. All studies in [17,18] led to the consistent conclusion that the FEM-FCT schemes are far better than the other approaches.…”

“…Thus, the solution of (11) can be expected to be much more expensive than the solution of the rest of the equations. The main goal of this paper consists in studying different approaches for discretizing (11) and exploring the impact of using expensive, higher order discretizations and inexpensive, low order methods on the accuracy and the computing times. The first approach which will be studied is the linear FEM-FCT scheme described above.…”

“…This section describes in detail the methods which were studied for solving the coupled system (6), (7), (8) for c A , (8) for c B , (10) and (11). The general strategy consists in first applying a temporal discretization with the same length of the time step to each equation of the system.…”