Explain the mean value filter in a bit more detail.

git-svn-id: https://svn.dealii.org/trunk@18706 0785d39b-7218-0410-832d-ea1e28bc413d

diff --git a/deal.II/examples/step-34/step-34.cc b/deal.II/examples/step-34/step-34.cc
index 5a81e76a485cbb0b28a783681d1fc4006e57b2e8..0c7920b56ba075b108f4fb8372b04502b2a03d79 100644 (file)
--- a/deal.II/examples/step-34/step-34.cc
+++ b/deal.II/examples/step-34/step-34.cc
@@ -1328,19 +1328,57 @@ void BEMProblem<dim>::assemble_system()
                                 // address this issue, we use two new
                                 // instruments of the library: the
                                 // MeanValueFilter class, and the
-                                // ProductMatrix class. The
-                                // MeanValueFilter has the interface
-                                // of a matrix (i.e. it has a
-                                // function MeanValueFilter::vmult),
-                                // with the effect that the output
-                                // vector equals the input vector
-                                // minus its mean value. We cascade
-                                // this operator with the system
-                                // matrix, and we obtain a matrix
-                                // whose result is renormalized to a
-                                // zero mean value Vector. In other
-                                // words, vectors that are multiplied
-                                // have mean value zero and therefore
+                                // ProductMatrix class.
+                                //
+                                // In essence, the idea is this: all
+                                // Krylov subspace solvers construct
+                                // an approximation the solution in
+                                // the space $\text{span}
+                                // \{b,Ab,A^2b,A^3b,\ldots,A^{n-1}b\}$
+                                // in the $n$-th iteration. We would
+                                // like the vectors in this space to
+                                // have mean value zero. To guarantee
+                                // this sort of thing, we should
+                                // instead consider the problem
+                                // $FAx=Fb$ where $F=I-\frac 1n
+                                // \mathbf{e}\mathbf{e}^T$ (with
+                                // $\mathbf e$ a vector of length $n$
+                                // with all entries equal to
+                                // one). $F$ is the matrix that given
+                                // a vector filters out its mean
+                                // value. The Krylov subspace that
+                                // GMRES constructs from this is
+                                // $\text{span}
+                                // \{Fb,FAb,FA^2b,FA^3b,\ldots,FA^{n-1}b\}$
+                                // (note here that $(FA)^k=FA^k$
+                                // because $A$ maps any vector $t$ to
+                                // exactly the same result as it
+                                // would map $Ft$ - that's the
+                                // definition of its kernel!). So
+                                // each of the elements of Krylov
+                                // subspace has mean value zero, and
+                                // as a consequence so does the
+                                // approximation $x^{(n)}$
+                                // constructed in the $n$-th
+                                // iteration.
+                                //
+                                // To implement this, we need a class
+                                // that represents the action of the
+                                // filter $F$. Sure enough, deal.II
+                                // has one of these: the
+                                // MeanValueFilter class has the
+                                // interface of a matrix (i.e. it has
+                                // a function
+                                // MeanValueFilter::vmult), with the
+                                // effect that the output vector
+                                // equals the input vector minus its
+                                // mean value. We cascade this
+                                // operator with the system matrix,
+                                // and we obtain a matrix $FA$ whose
+                                // result is renormalized to a zero
+                                // mean value vector. In other words,
+                                // vectors that are multiplied have
+                                // mean value zero and therefore
                                 // never feel the fact that the
                                 // system matrix has a kernel for
                                 // these. The combined matrix object