History for Libmesh Boundary Conditions

added:
  Section 2.4 Boundary Conditions of Ondrej Certik's "notes on
  FEM":http://ondrej.certik.cz/libmesh/fem.ps describes the penalty method used
  to apply Dirichlet boundary conditions.

  * The boundary fluxes for the conservation equations corresponding to test
    functions that are non-zero on the Dirichlet boundary are not known.  Zero
    appears to be substituted for the fluxes, making the equations incorrect.

  * The L2 projection of the Dirichlet boundary value errors are added to the
    matrix, "multiplied by some large factor so that, in floating point
    arithmetic, the existing (smaller) entries in the matrix and
    right-hand-side are effectively ignored" ("Example
    3":http://libmesh.sourceforge.net/ex3.php) and so the incorrect
    conservation equations are effectively ignored.

  * These issues are discussed in "this thread":http://sourceforge.net/mailarchive/message.php?msg_id=37035937.

  The Nitsche method for Dirichlet boundary conditions is similar to the
  penalty method but corrects the conservation equations so that they are
  consistent.

  * There is still a parameter to be selected for the Dirichlet terms that
    depends on the mesh, but it does not need to be so large as to swamp the
    conservation equations and so the system is better conditioned.
  
  * More details are in M. Juntunen and R. Stenberg's "A finite element method
    for general boundary
    conditions":http://math.tkk.fi/~rstenber/Publications/nscm_general_boundary.pdf
    for the Proceedings of the 18 Nordic Seminar on Computational Mechanics,
    which also points out the inconsistency of the penalty method.

  * The Nitsche method can also be used on interfaces between portions of the
    domain with non-matching meshs, as analysed in R. Becker, P. Hansbo, and
    R. Stenberg's "A finite element method for domain decomposition with
    non-matching grids":http://www.math.hut.fi/~rstenber/Publications/Becker-Hansbo-Stenberg.pdf in Mathematical Modelling and Numerical Analysis **37**
    (2003) 209-225.

  "This message from Ben
  Kirk":http://sourceforge.net/mailarchive/message.php?msg_id=8617660 says
  "there is a more "conventional" way to assign Dirichlet BCs in
   the case of lagrange elements..."

  * DenseMatrix::condense can be used to do this.  Some of the issues involved
    are mentioned in "this
    thread":http://sourceforge.net/mailarchive/message.php?msg_id=15191940.

  * It may be possible to use 'DofMap::add_constraint_row' to add a zero
    contraint to remove equations corresponding to Dirichlet boundaries.
    Corrections would need to be made to the solution and right hand side for
    non-homogeneous Dirichlet conditions.  "This
    message":http://sourceforge.net/mailarchive/message.php?msg_id=11202052
    uses add_constraint_row for periodic boundary conditions (but check
    followups for suggested improvements).

  "This message":http://sourceforge.net/mailarchive/message.php?msg_id=9144809
  and "this
  message":http://sourceforge.net/mailarchive/message.php?msg_id=7524357 discuss
  'BoundaryMesh' and 'Mesh::boundary_info'.

  * I (Travis) have created an example that implements Dirichlet boundary conditions in
    the more natural (to me) way.  Rows of the matrix corresponding to Dirichlet points
    are replaced by identity equations.  The other rows of the matrix are then changed to
    make the matrix symmetric.  Furthermore, the right-hand side incorporates the effect
    of boundary conditions.