From: Manuel Kessler Subject: Ladyshenskaya's constant Date: 1 Feb 2000 20:49:28 +0100 Newsgroups: sci.math.num-analysis Summary: [missing] Keywords: Ladyshenskaya, Babuska-Brezzi condition Hello Numerical Analysts, in the last stages of my PhD thesis I'd like to ask some competent people to make me aware of any missed papers, proofs and other thougts concerning Ladyshenskaya's condition and the Babuska-Brezzi condition (it's discrete counterpart). The former one can be stated as the equivalence between the L^2 norm of a function and the H^-1 norm of its gradient, with Ladyshenskaya's constant as the lower equivalence factor (1 is the upper one). I know already the equivalence to Korn's and Friedrichs' inequality in two dimensions due to the work af Horgan, Payne, Weinberger and Oleinik. There I found also some quite direct proofs of Korn's and Friedrichs' inequality in two dimensions, where the constant can be estimated by means of properties of the domain under consideration, especially in the latter case. The equivalence is based on the equivalence of some eigenproblems related to each inequality and correspondance of eigenfunctions and eigenvalues. However, no such equivalence seems to be known for three and more dimensions. Furthermore, the existence of Ladyshenskaya's constant and hence unique solvability of the Stokes system can be proved for domains consiting of a finite number of starshaped domains with respect to a ball. It is known that the condition fails for domains with an external cusp or a cusp-like edge. However, all such proofs are either connected to the Korn problem (Babuska-Aziz, Horgan/Payne/Weinberger) and thus restricted to two dimensions or use the existence in the half space and construct a velocity from the pressure satisfying div v=p by partitioning the domain and connecting the half space solutions (Galdi, Duvaut/Lions). The latter technique is quite involved and thus the dependence of the constant from the domain can't be tracked anymore along the proof. I'm also interested in information about the discrete counterpart of Ladyshanskaya's condition, the Babuska-Brezzi inequality. Here again such a constant B is required for convergence of the discrete solution to the true continous one. I know about the importance of respecting this inequality when choosing Finite Element spaces for e.g. the Stokes problem, and I know several proofs for several stable elements, e.g. the Mini element, the Taylor-Hood element (continous piecewise linear pressure, continous piecewise quadratic velocity or continous piecewise linear velocity on a once refined mesh) and the nonconforming Crouzeix-Raviart element. All proofs for those elements require the existence of the continous constant, which is a sensible thing since it is kind of an upper bound for the discrete one (at least in the limit h->0). However, tracking constants is not easy, since several interpolation operators occur with unknown or at least difficult to compute approximation errors (in the sense of sharp bounds) for nontrivial triangulations and therefore no sharp correspondence between discrete and continous constant seems to be provable. So, to make my long story come to a short end: - Do you know of any results which disagree with my abovementioned thoughts? - Are there any analytical results for domains other than the circle, ellipsis and the circular ring in 2D, the sphere (and ellipsoid) in 3D, especially maybe the square and/or cubus? - Do you know of any recent work done on this problem and/or proofs simple enough to follow the evolution of the constant? Many thanks for any and all comments. Ciao, Manuel For those people which may be competent, but are not fluent in my terminology, here a short summary of the problem: When solving partial differential equations with constraints in a domain, e.g. the Stokes equations as the canonical example, existence and uniqueness of the solution require the existence of a constant L satisfying (p,div v) inf sup ----------- > L > 0 p v |Dv| |p| where p is a square-integrable function (L^2) and v a vector-valued function of the Sobolev space H^1 with zero boundary conditions. (p,div v) denotes the usual scalar product in L^2, |Dv| is the H^1 semi-norm (equivalent to the full norm due to the zero boundary condition and Poincare's inequality) and finally |p| is the usual L^2 norm. This is called Ladyshenskaya's condition or inequality and the constant L, which is a function of the domain, is Ladyshenskaya's constant. The formally identical inequality, but this time using discrete spaces for p and v, most notably Finite Element spaces coming from e.g. continous piecewise linear functions on a triangulation of the domain, is named Babuska-Brezzi condition, and is very important in the convergence analysis of discretizations of said partial differential equations. This leads to using different spaces for v and p, since otherwise non-convergent highly oscillatory functions (so-called checkerboard modes) in the pressure p appear. A similar effect in the computation of elastostatic problems is called locking. ------------------------------------------------------------------------------ Manuel Kessler PhD Student at the University of Wuerzburg, Germany, Mathematics Department SNAIL: Stra"sen"acker 60, D-71634 Ludwigsburg, Germany EMAIL: mlkessle@cip.physik.uni-wuerzburg.de WWW: http://cip.physik.uni-wuerzburg.de/~mlkessle