comment on spurious eigenvalues in Step-36

diff --git a/examples/step-36/doc/intro.dox b/examples/step-36/doc/intro.dox
index f0eb3867cbf887cbc4a0606dd674141691d94557..32f9a74845c3d5b4413d9adb7e03f59035c2d8fa 100644 (file)
--- a/examples/step-36/doc/intro.dox
+++ b/examples/step-36/doc/intro.dox
@@ -93,6 +93,22 @@ $A$ and $M$ that we will solve in the current tutorial program. We
 will want to solve it for the lowermost few eigenvalue/eigenfunction
 pairs.
 
+Zero Dirichlet boundary conditions result in some degrees of freedom (DoFs) being constrained. 
+Those DoFs will not be eliminated from the algebraic system. 
+Rather, during assembly process 
+the condensed matrices $A$ and $M$ are made to have zero rows and columns for the corresponding 
+DoFs and non-zero diagonal elements. 
+For the local matrices these diagonal elements, if non-zero, 
+are set to their absolute value. Otherwise, they are set to the average of  
+absolute values of the diagonal. 
+This results in a block-diagonal structure of $A$ and $M$, where one block
+corresponds to the eigenpairs of interest and the other introduces 
+spurious (real) positive eigenvalues $\varepsilon_{h,i}=A_{ii}/M_{ii}$ (where $i$ are constrained DoFs). 
+In order to filter out the spurious eigenvalues, one could scale the diagonal elements of either matrix 
+thus shifting them away from the frequency of interest in the eigen-spectrum. 
+However, this strategy was not pursued here as those eigenvalues happen to be greater than the lowest 
+five that we will calculate.
+
 <h3>Implementation details</h3>
 
 The program below is essentially just a slightly modified version of 

diff --git a/examples/step-36/step-36.cc b/examples/step-36/step-36.cc
index acf41d7fa265f400e723868ae9c264cb0cf1668d..4b5c78c59b50a97897b655934402b64923449a76 100644 (file)
--- a/examples/step-36/step-36.cc
+++ b/examples/step-36/step-36.cc
@@ -292,6 +292,28 @@ namespace Step36
     // be compressed as no more entries will be added:
     stiffness_matrix.compress (VectorOperation::add);
     mass_matrix.compress (VectorOperation::add);
+
+
+    // Before leaving the function,
+    // we calculate spurious eigenvalues,
+    // introduced to the system by zero Dirichlet constraints.
+    double min_ev = 1e+10,
+          max_ev = -min_ev;
+
+    for (unsigned int ind = 0; ind < dof_handler.n_dofs(); ind++)
+      if (constraints.is_constrained(ind))
+       {
+         const double ev = stiffness_matrix(ind,ind)/mass_matrix(ind,ind);
+         if ( min_ev > ev )
+           min_ev = ev;
+         if ( max_ev < ev )
+           max_ev = ev;
+       }
+
+    std::cout << "   Spurious eigenvalues are in "
+       << "["<<min_ev<<":"<<max_ev<<"]"
+       << std::endl;
+
   }