Hello,
Could anyone recommend to me literature on how to numerically investigate coercivity of a bilinear form? I am interested in bilinear form from the discontinuous Galerkin method. In other words how to numerically test that A(v,v) >= C||v|| for all functions v in the DG space?
Thanks for reading!
If it’s a DG discretisation of an elliptic problem that you’re looking at, you can simply modify the value of the penalty parameter. But this obviously depends on your choices of numerical flux. See, e.g., the ridiculously well cited review.
Thank you. Do you mean to investigate the linear system convergence when modify the penalty parameter? If so, this is partly what caused my interest in this problem. Because using the direct solver (umfpack) works well also when the penalty parameter is very small.
I tested this on simple SIPG problem on unit square, f=1 in right hand side, u=0 on boundary. Solution converges with penalty parameter=0, but it clear visible wrong. With penalty parameter=1e-13 solution looks almost same as with penalty parameter=10.
I think this is not new and stability of method is confused with good linear system solver. For my question I am interested in numerically investigate stability of DG without stability of linear system solver.
Thank you very much!
With no penalty parameter you may just be solving a system equivalent to the Baumann-Oden flux for the elliptic problem which is (I believe, but check this to be sure since my memory is fuzzy) only coercive when measured in a semi-norm. So the Baumann-Oden DG FEM discretisation of the weak formulation may be stable, but not coercive, giving you a suboptimal approximation. See the following figure which compares the convergence rates of the error of a solution approximation computed from a linear Poisson problem for various DG flux formulations. Reproduced from here.
I also remind you that I’m writing this from vague memories. I strongly suggest you read the literature to be sure.
Thank you. I understand. But I think it can be interesting to test coercivity by numerical investigation.
Suppose one have DG method for problem and theory is not known or not known to engineer.
Testing convergence is very good, then engineer can know method is ok. It will guarantee good method in range of h tested.
Test of coercivity, maybe can complement test convergence. Idea: Compute approximate constant in A(v,v) >= C||v||. If C very large, engineer is safe. If C small, then need more investigation such as convergence investigation.
Expensive investigation for finding C is maybe ok. Must be compared with cost of mesh refine and testing convergence, if this can be avoided.
Perhaps eigenvalue problem of A? Minimum eigenvalue?
Thank you for reading!
I’m assuming your A is symmetric.
If yes, you can compute its eigenvalues and if they’re all strictly positive, then your problem is coercive.