Stabilization parameters

Finite Element Methods (FEM) are widely used to approximate the solutions of partial differential equations (PDEs) in a variety of fields, including fluid dynamics, solid mechanics, and heat transfer. However, when applied to certain classes of problems — particularly those involving advection-dominated flows, incompressible materials, or nearly singular systems — the standard Galerkin formulation can lead to numerical instabilities such as spurious oscillations, locking, or non-physical solutions.

To address these issues, stabilization techniques are introduced. These typically involve adding carefully designed terms to the weak form of the governing equations, weighted by stabilization parameters. These parameters control the amount of artificial diffusion, penalization, or residual-based correction applied locally in the computation.

Some common contexts where stabilization is essential:

• Advection-dominated flows: where high Péclet or Reynolds numbers can cause oscillations if the numerical diffusion is insufficient.
• Incompressible materials or fluids: where enforcing divergence-free conditions can lead to pressure instabilities or checkerboard patterns without pressure stabilization.
• Mixed finite element formulations: where satisfying the Ladyzhenskaya–Babuška–Brezzi (LBB) condition may require stabilization to avoid locking or instability.

Stabilization parameters are typically problem-dependent, often involving local mesh size, material properties, or flow velocities, and their careful tuning is crucial for balancing stability and accuracy.

Popular examples include:

• SUPG (Streamline Upwind Petrov–Galerkin) stabilization for advection-dominated flows.
• PSPG (Pressure Stabilizing Petrov–Galerkin) for incompressible Navier–Stokes equations.
• Galerkin/Least-Squares (GLS) methods.
• Penalty methods for enforcing incompressibility or interface conditions.

Gales stabilization parameters

Gales derives its name from the Galerking Least Squares stabilization tecnique, originally proposed by Hughes, France, Hauke and collaborators starting from the early 1986s.

The foundational paper typically credited for formalizing the Galerkin/Least-Squares (GLS) stabilization method is:

Hughes, T.J.R., Franca, L.P., & Balestra, M. (1986)
A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška–Brezzi condition: A stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations
Computer Methods in Applied Mechanics and Engineering, 59(1), 85–99.
https://doi.org/10.1016/0045-7825(86)90127-9

This paper introduced a Petrov-Galerkin formulation (which evolved into GLS) for the Stokes problem, using a least-squares approach to stabilize pressure-velocity coupling when equal-order interpolations were used — effectively circumventing the Babuška–Brezzi (inf-sup) condition.

Later, the method was extended to the Navier–Stokes equations and other PDE systems, with Hughes and Franca continuing to develop the theoretical framework in several influential papers through the late 1980s and early 1990s.

Further, the paper:

Hughes, T.J.R., Franca, L.P., & Mallet, M. (1987)
A new finite element formulation for computational fluid dynamics: VII. The Galerkin/least-squares method for advective-diffusive equations.
Computer Methods in Applied Mechanics and Engineering, 65(1), 109–126.
https://doi.org/10.1016/0045-7825(87)90173-8

extended the GLS formulation to advection-diffusion problems, addressing numerical instabilities in convection-dominated flows.

The framework of GLS that is the base of Gales was developed in:

Shakib, F, Hughes, T.J.R., Johan, Z. (1991)
A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier-Stokes equations.
Computer Methods in Applied Mechanics and Engineering, 89, 141–219.
https://doi.org/10.1016/0045-7825(91)90041-4

The paper introduced the tau matrix of GLS and Discontinuity Capturing operator for the compressible Navier-Stokes equations in entropy variables.

Hauke & Hughes extended the formulation to pressure primitive variables to handle from compressible to incompressible flows in a unified way.

Hauke, G., Hughes, T.J.R. (1998)
A comparative study of different sets of variables for solving compressible and incompressible flows .
Computer Methods in Applied Mechanics and Engineering, 153 (1-2), 1–44.
https://doi.org/10.1016/S0045-7825(97)00043-1

From there on several versions of tau matrices and discontinuity capturing operators were developed in literature.

Gales makes use of the following tau matrices and discontinuity capturing operators:

• 2001 tau diagonal for compressible flows, Hauke (2001)
• 2001 tau non diagonal for compressible flows, Hauke (2001)
• 2019 tau non diagonal for compressible flows, Hauke (2001)
• 2007 tau diagonal for incompressible flows, Bazilevs et al. (2007)
• 2014 tau diagonal, Trelles, J. P., & Modirkhazeni, S. M. (2014)
• 2005 tau, Polner, M. (2005)
• 1998 discontinuity capturing operator, Hauke & Hughes (1998)
• 2006 discontinuity capturing operator, Tezduyar, T. E., & Senga, M. (2006)
• GSL for heat conduction, F. Ilinca & J.-F. Heetu (2002)

References

Hauke, G. (2001). Simple stabilizing matrices for the computation of compressible flows in primitive variables. Computer Methods in Applied Mechanics and Engineering, 190(51-52), 6881-6893. https://doi.org/10.1016/S0045-7825(01)00267-5

Bazilevs et al. (2007). Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput. Methods Appl. Mech. Engrg. 197 (2007) 173–201. doi:10.1016/j.cma.2007.07.016

Trelles, J. P., & Modirkhazeni, S. M. (2014). Variational multiscale method for nonequilibrium plasma flows. Computer Methods in Applied Mechanics and Engineering, 282, 87-131. https://doi.org/10.1016/j.cma.2014.09.001

Tezduyar, T. E., & Senga, M. (2006). Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Computer Methods in Applied Mechanics and Engineering, 195(13-16), 1621-1632. https://doi.org/10.1016/j.cma.2005.05.032

F. Ilinca & J.-F. Heetu (2002) Galerkin gradient least-squares formulations for transient conduction heat transfer. Computer Methods in Applied Mechanics and Engineering, 191(13-16), 3073-3097. https://doi.org/10.1016/S0045-7825(02)00242-6