From f0e9dcbc1ac59a8581604443e162afe969f23bf4 Mon Sep 17 00:00:00 2001
From: Kilian <kilian.weishaupt@iws.uni-stuttgart.de>
Date: Mon, 6 Apr 2020 19:50:46 +0200
Subject: [PATCH] [linear][uzawa] Improve docu
---
doc/handbook/dumux-handbook.bib | 25 ++++++++++++++++
dumux/linear/preconditioners.hh | 53 +++++++++++++++++++++++++++++++++
2 files changed, 78 insertions(+)
diff --git a/doc/handbook/dumux-handbook.bib b/doc/handbook/dumux-handbook.bib
index 88860ce45f..9b5588c9ac 100644
--- a/doc/handbook/dumux-handbook.bib
+++ b/doc/handbook/dumux-handbook.bib
@@ -1882,3 +1882,28 @@ year={1999}
pages={37},
year={1964}
}
+
+@article{benzi2005,
+ doi = {10.1017/s0962492904000212},
+ url = {https://doi.org/10.1017/s0962492904000212},
+ year = {2005},
+ publisher = {Cambridge University Press ({CUP})},
+ volume = {14},
+ pages = {1--137},
+ author = {Michele Benzi and Gene H. Golub and J\"{o}rg Liesen},
+ title = {Numerical solution of saddle point problems},
+ journal = {Acta Numerica}
+}
+
+@article{ho2017,
+ doi = {10.1137/16m1076770},
+ url = {https://doi.org/10.1137/16m1076770},
+ year = {2017},
+ publisher = {Society for Industrial {\&} Applied Mathematics ({SIAM})},
+ volume = {39},
+ number = {5},
+ pages = {S461--S476},
+ author = {Nguyenho Ho and Sarah D. Olson and Homer F. Walker},
+ title = {Accelerating the Uzawa Algorithm},
+ journal = {{SIAM} Journal on Scientific Computing}
+}
diff --git a/dumux/linear/preconditioners.hh b/dumux/linear/preconditioners.hh
index 346c3dfc71..16df85cfba 100644
--- a/dumux/linear/preconditioners.hh
+++ b/dumux/linear/preconditioners.hh
@@ -44,6 +44,40 @@
namespace Dumux {
#if DUNE_VERSION_GTE(DUNE_ISTL,2,7)
+/*!
+ * \ingroup Linear
+ * \brief A preconditioner based on the Uzawa algorithm for saddle-point problems of the form
+ * \f$
+ \begin{pmatrix}
+ A & B \\
+ C & D
+ \end{pmatrix}
+
+ \begin{pmatrix}
+ u\\
+ p
+ \end{pmatrix}
+
+ =
+
+ \begin{pmatrix}
+ f\\
+ g
+ \end{pmatrix}
+ * \f$
+ *
+ * This preconditioner is especially suited for solving the incompressible (Navier-)Stokes equations.
+ * Here, \f$D = 0\f$ and \f$B = C^T\f$ if \f$\rho = 1\f$.
+ * We do not expect good convergence if energy or mass transport is considered.
+ *
+ * See: Benzi, M., Golub, G. H., & Liesen, J. (2005). Numerical solution of saddle point problems. Acta numerica, 14, 1-137 \cite benzi2005 and <BR>
+ * Ho, N., Olson, S. D., & Walker, H. F. (2017). Accelerating the Uzawa algorithm. SIAM Journal on Scientific Computing, 39(5), S461-S476 \cite ho2017
+ *
+ * \tparam M Type of the matrix.
+ * \tparam X Type of the update.
+ * \tparam Y Type of the defect.
+ * \tparam l Preconditioner block level (for compatibility reasons, unused).
+ */
template<class M, class X, class Y, int l = 1>
class SeqUzawa : public Dune::Preconditioner<X,Y>
{
@@ -189,6 +223,25 @@ private:
#endif
}
+ /*!
+ * \brief Estimate the relaxation factor omega
+ *
+ * The optimal relaxation factor is omega = 2/(lambdaMin + lambdaMax), where lambdaMin and lambdaMax are the smallest and largest
+ * eigenvalues of the Schur complement -C*Ainv*B (assuming D = 0).
+ * lambdaMax can be easily determined using the power iteration algorithm (https://en.wikipedia.org/wiki/Power_iteration) and lambdaMin
+ * could be estimated in a similar manner. We do not consider lambdaMin because for certain cases, e.g., when C contains some rows of zeroes only,
+ * this estimate will fail.
+ *
+ * Instead we assume that lambdaMin is sufficiently close to lambdaMax such that omega = 1/lambdaMax.
+ * This seems to work rather well for various applications.
+ * We will underestimate omega by a factor of 2 in the worst case (i.e, lambdaMin = 0).
+ *
+ * When facing convergence issues, you may set LinearSolver.Preconditioner.Verbosity = 1 to see the estimate of lambdaMax.
+ * In a new simulation run, you can then set LinearSolver.Preconditioner.DetermineRelaxationFactor = false and set some other value
+ * for LinearSolver.Preconditioner.Relaxation based on the estimate of lambdaMax.
+ *
+ * See: Benzi, M., Golub, G. H., & Liesen, J. (2005). Numerical solution of saddle point problems. Acta numerica, 14, 1-137.
+ */
scalar_field_type estimateOmega_() const
{
using namespace Dune::Indices;
--
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