Observations on degenerate saddle point problems

Details Observations on degenerate saddle point problems Observations on degenerate saddle point problems

2007-04-09

arxiv.org/abs/0704.1066v1

Comput. Methods Appl. Mech.Engrg. 196, Issues 37-40, 3742-3749 (2007)

Mathematics

Numerical Analysis

8 pages

Scientific paper

10.1016/j.cma.2006.10.019

We investigate degenerate saddle point problems, which can be viewed as limit cases of standard mixed formulations of symmetric problems with large jumps in coefficients. We prove that they are well-posed in a standard norm despite the degeneracy. By wellposedness we mean a stable dependence of the solution on the right-hand side. A known approach of splitting the saddle point problem into separate equations for the primary unknown and for the Lagrange multiplier is used. We revisit the traditional Ladygenskaya--Babu\v{s}ka--Brezzi (LBB) or inf--sup condition as well as the standard coercivity condition, and analyze how they are affected by the degeneracy of the corresponding bilinear forms. We suggest and discuss generalized conditions that cover the degenerate case. The LBB or inf--sup condition is necessary and sufficient for wellposedness of the problem with respect to the Lagrange multiplier under some assumptions. The generalized coercivity condition is necessary and sufficient for wellposedness of the problem with respect to the primary unknown under some other assumptions. We connect the generalized coercivity condition to the positiveness of the minimum gap of relevant subspaces, and propose several equivalent expressions for the minimum gap. Our results provide a foundation for research on uniform wellposedness of mixed formulations of symmetric problems with large jumps in coefficients in a standard norm, independent of the jumps. Such problems appear, e.g., in numerical simulations of composite materials made of components with contrasting properties.

Affiliated with

Also associated with

No associations

Rating

Observations on degenerate saddle point problems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Observations on degenerate saddle point problems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Observations on degenerate saddle point problems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-387193