Discrete Stability, DPG Method and Least Squares
Leszek F. Demkowicz,
(University of Texas, Austin, USA)
Martes 13 de diciembre 2011
Ever since the ground breaking paper of Ivo Babuska [1], everybody from
Finite Element (FE) community has learned the famous phrase:
``Discrete stability and approximability imply convergence''
In fact, Ivo gets only a partial credit for the phrase that had already been
used for some time by the Finite Difference (FD) people since an equally
well known result of Peter Lax [2]. The challenge in establishing convergence
comes from the fact that, except for a relative small class of ``safe''
coercive (elliptic) problems, continuous stability DOES NOT imply discrete stability.
In other words, the problem of interest may be well posed at the continuous
level but this does not imply that the corresponding FD or FE discretization
will automatically converge. No wonder then that the FE numerical analysis
community spent the last 40+ years coming up with different ideas how to
generate discretely stable schemes coming up with such famous results as
Mikhlin's theory of asymptotic stability for compact perturbations of
coercive problems, Brezzi's theory for problems with constraints, concept
of stabilized methods starting with SUPG method of Tom Hughes, the bubble
methods, stabilization through least-squares, stabilization through a proper
choice of numerical flux including a huge family of DG methods starting with
the method of Cockburn and Shu, and a more recent use of exact sequences.
In the first part of my presentation I will recall Babuska's Theorem and
review shortly the milestones in designing various discretely stable
methods listed above.
In the second part of my presentation, I will present the Discontinuous
Petrov-Galerkin method developed recently by Jay Gopalakrishan and myself [3,4].
The main idea of the method is to employ (approximate) optimal test functions
that are computed on the fly at the element level using Bubnov-Galerkin
and an enriched space. If the error in approximating the optimal test
functions is negligible, the method AUTOMATICALLY guarantees the discrete
stability, provided the continuous problem is well posed. And this holds
for ANY linear problem. The result is shocking until one realizes that
we are working with a unconventional least squares method. The twist lies
in the fact that the residual lives in a dual space and it is computed
using dual norms.
The method turns out to be especially suited for singular perturbation problems
where one strives not only for stability but also for ROBUSTNESS, i.e. a stability
UNIFORM with respect to the perturbation problem. I will use to important
model problems: convection-dominated diffusion and linear acoustics equations
to illustrate the main points.
[1] I. Babuska, Error-bounds for Finite Element Method. Numer. Math, 16, 1970/1971.
[2] P. Lax, Numerical Solution of Partial Differential Equations. Amer. Math. Monthly,
72 1965 no. 2, part II.
[3] L. Demkowicz and J. Gopalakrishnan. A Class of Discontinuous Petrov-Galerkin Methods.
Part II: Optimal Test Functions. Numer. Meth. Part. D. E., 27, 70-105, 2011.
[4] L. Demkowicz and J. Gopalakrishnan. A New Paradigm for Discretizing Difficult Problems:
Discontinuous Petrov Galerkin Method with Optimal Test Functions. Expressions (publication
of International Association for Computational Mechanics), November 2010.