From: Daniel Arndt
Date: Mon, 1 Aug 2016 17:40:35 +0000 (+0200)
Subject: Add literature to step-35
X-Git-Tag: v8.5.0-rc1~814^2
X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=a7578047bb2008a3d867256188165bf76c5836bd;p=dealii.git
Add literature to step-35
---
diff --git a/examples/step-35/doc/intro.dox b/examples/step-35/doc/intro.dox
index f5fe3d6490..3685f54be8 100644
--- a/examples/step-35/doc/intro.dox
+++ b/examples/step-35/doc/intro.dox
@@ -108,8 +108,7 @@ $(u^0,p^0,\phi^0=0)$ and $(u^1,p^1,\phi^1=p^1-p^0)$. Then the projection method
-Without going into details, which can be found in the corresponding literature, let us remark a few things about the
-projection methods that we have just described:
+Without going into details, let us remark a few things about the projection methods that we have just described:
- The advection term $u\cdot\nabla u$ is replaced by its skew symmetric form
@f[
@@ -143,6 +142,35 @@ projection methods that we have just described:
+
+The standard incremental scheme and the rotational incremental scheme were first considered by van Kan in
+
+ - J. van Kan, "A second-order accurate pressure-correction scheme for viscous incompressible flow",
+ SIAM Journal on Scientific and Statistical Computing, vol. 7, no. 3, pp. 870â891, 1986
+
+and is analyzed by Guermond in
+
+ - J.-L. Guermond, "Un résultat de convergence dâordre deux en temps pour
+ lâapproximation des équations de NavierâStokes par une technique de projection incrémentale",
+ ESAIM: Mathematical Modelling and Numerical Analysis, vol. 33, no. 1, pp. 169â189, 1999
+
+for the case $\nu = 1$.
+It turns out that this technique suffers from unphysical boundary conditions for the kinematic pressure that
+lead to reduced rates of convergence. To prevent this, Timmermans et al. proposed in
+
+ - L. Timmermans, P. Minev, and F. Van De Vosse,
+ "An approximate projection scheme for incompressible flow using spectral elements",
+ International Journal for Numerical Methods in Fluids, vol. 22, no. 7, pp. 673â688, 1996
+
+the rotational pressure-correction projection method that uses a divergence correction for the kinematic pressure.
+A thorough analysis for scheme has first been performed in
+
+ - J.-L. Guermond and J. Shen, "On the error estimates for the rotational pressure-correction projection methods",
+ Mathematics of Computation, vol. 73, no. 248, pp. 1719â1737, 2004
+
+for the Stokes problem.
+
+
The Fully Discrete Setting
To obtain a fully discrete setting of the method we, as always, need a variational formulation. There is one