For many problem settings, finite element methods based on Galerkin variational principles do not allow for the independent choice of the finite element spaces for the discretization of different unknowns. For example, in practice, for the Stokes or Navier-Stokes systems, one cannot choose discretization spaces for the velocity and pressure based on the same grid and same degree polynomials. It is well known that the spaces have to satisfy an inf-sup or LBB (Ladyzhenskaya-Babuska-Brezzi) condition. In principle, this constraint on the finite element spaces is no longer considered to pose any difficulties because, in many settings, the inf-sup condition is well understood theoretically and many classes of finite element pairs have been constructed that satisfy the condition. However, in practice, one often would like to use the same grid and/or same degree polynomials for all variables, not just because of ease of programming, but also because a code, e.g., a Navier-Stokes code, often has to interface with other codes that model, e.g., thermal, transport, or chemical effects.
Other settings for which straightforward discretization of Galerkin principles are not effective in practice are typified by problems for which convection dominates diffusion, e.g., high-Reynolds number flows, or there is no diffusion at all, e.g., hyperbolic conservation laws. In the first case, computing stable approximation from mixed-Galerkin methods requires grid sizes that are too small to be of practical use. In the second case, stabilization is needed at any grid size to remove non-physical Gibbs oscillations from the approximate solution.
Stabilized finite element methods are based on modifications of variational principles (usually Galerkin principles) such that finite element discretizations no longer require the imposition of an inf-sup condition or which allow for the stable solution of, e.g., convection-dominated problems. Consistently stabilized methods refer to stabilized methods such that the exact solution of the PDE system satisfies the modified system as well. For inf-sup stabilization, the are constructed to allow for the use of the same grid and the same degree polynomials for all variables. In the convection stabilization case, they are constructed to allow for solutions at practical grid sizes and time steps.
Our research involves both inf-sup stabilization and stabilization of convection dominated problems. We have developed and analyzed new algorithms or enhanced versions of existing algorithms and have also addressed may issues that arise in when stabilized methods are implemented.
T. Barth, P. Bochev, M. Gunzburger, and J. Shadid; A taxonomy of consistently stabilized finite element methods for the Stokes problem, SIAM J. Sci. Comp. 25 2004, 1585-1607.
P. Bochev, M. Gunzburger, and J. Shadid; On inf-sup stabilized finite element methods for transient problems, Comp. Meth. Appl. Mech. Engrg. 193 2004, 1471-1489.
P. Bochev, M. Gunzburger, and J. Shadid; Stability of the SUPG finite elements for transient advection-diffusion problems, Comp. Meth. Appl. Mech. Engrg. 193 2004, 2301-2323.
P. Bochev and M. Gunzburger; An absolutely stable pressure-Poisson stabilized finite element method for the Stokes equations, SIAM J. Numer. Anal. 42 2004, 1189-1207.
P. Bochev and M. Gunzburger; On least-squares finite element methods for the Poisson equation and their connection to the Dirichlet and Kelvin principles, SIAM J. Numer. Anal. 43 2005, 340-362.
P. Bochev, C. Dohrmann, and M. Gunzburger; Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal. 44 2006, 82-101.
P. Bochev, M. Gunzburger. and R. Lehoucq; On stabilized finite element methods for the Stokes problem in the small time-step limit, Int. J. Numer. Meth. Fluids 53 2007, 573-597.
Last updated: 12/29/09 by Max Gunzburger