Also update some commentary.

diff --git a/examples/step-57/step-57.cc b/examples/step-57/step-57.cc
index a282ea66b5a594345d7543ab016e329104d6aeb6..d8f139e0eea3639e0cfc8685ce886d3d7b8ceaa4 100644 (file)
--- a/examples/step-57/step-57.cc
+++ b/examples/step-57/step-57.cc
@@ -487,18 +487,32 @@ namespace Step57
           }
       }
 
+    // If we were asked to assemble the Newton matrix, then we also built
+    // a pressure mass matrix in the bottom right block of the matrix.
+    // We only need this for the preconditioner, so we need to copy it
+    // in into a separate matrix object, followed by zeroing out this
+    // block in the Newton matrix.
+    //
+    // Note that settings this bottom right block to zero is not identical to
+    // not assembling anything in this block, because applying boundary values
+    // and hanging node constraints (in the
+    // `constraints_used.distribute_local_to_global()` call above) puts entries
+    // into this block. As a consequence, setting the $(1,1)$ block to zero
+    // below does not result in what would have happened if we had just not
+    // assembled a pressure mass matrix in that block to begin with.
+    //
+    // The difference is that if we had not assembled anything in this block,
+    // dealing with constraint degrees of freedom would have put entries on
+    // the diagonal of the $(1,1)$ block whereas the last operation below,
+    // zeroing out the entire block, results in a system matrix with
+    // rows and columns that are completely empty. In other words, the
+    // linear problem is singular. Luckily, however, the FGMRES solver we
+    // use appears to handle these rows and columns without any problem.
     if (assemble_matrix)
       {
-        // Finally we move pressure mass matrix into a separate matrix:
         pressure_mass_matrix.reinit(sparsity_pattern.block(1, 1));
         pressure_mass_matrix.copy_from(system_matrix.block(1, 1));
 
-        // Note that settings this pressure block to zero is not identical to
-        // not assembling anything in this block, because this operation here
-        // will (incorrectly) delete diagonal entries that come in from
-        // hanging node constraints for pressure DoFs. This means that our
-        // whole system matrix will have rows that are completely
-        // zero. Luckily, FGMRES handles these rows without any problem.
         system_matrix.block(1, 1) = 0;
       }
   }