From a56d4992926f8f7b2dd88a1282de950bd22a81bd Mon Sep 17 00:00:00 2001 From: Juha Jeronen Date: Tue, 15 Mar 2022 10:46:38 +0200 Subject: [PATCH] add comment on Brenner & Scott's remarks on SIPG --- demo/poisson_dg.py | 16 ++++++++++++++++ 1 file changed, 16 insertions(+) diff --git a/demo/poisson_dg.py b/demo/poisson_dg.py index 80b8b3f..abeadab 100644 --- a/demo/poisson_dg.py +++ b/demo/poisson_dg.py @@ -111,6 +111,22 @@ def boundary(x): # https://scicomp.stackexchange.com/questions/20078/matlab-implementation-of-2d-interior-penalty-discontinuous-galerkin-poisson-prob # https://fenicsproject.org/qa/3974/expression-in-interior-facets/ # +# +# Also see Brenner & Scott, sec. 10.5. Reading between the lines, the motivation of the +# stabilization term seems to be that it is needed to set up a suitable energy norm +# from which one can prove coercivity, whence stability. +# +# The authors also remark (p. 291) that for a consistent method, convergence only +# depends on stability, which in turn involves boundedness (i.e. continuity in the +# bilinear form sense) and coercivity of the relevant bilinear forms. It turns out that +# there is a critical value for the stabilization parameter; the parameter must be +# critical or larger for the discrete bilinear form to be coercive. See lemma 10.5.19. +# +# Reference: +# Susanne C. Brenner & L. Ridgway Scott. 2010. The Mathematical Theory +# of Finite Element Methods. 3rd edition. Springer. ISBN 978-1-4419-2611-1. +# +# # TL;DR summary: in a consistent formulation, we can arbitrarily add: # - Terms with [u] on the interior facets # - Terms with (u - u0) on the Dirichlet boundary facets