Benchmark solution
=====================


Calculation method
-----------------

The analytical reference solution is given by:


.. math::
 :label: eq-1

    T\ mathrm {=} 100\ mathrm {\ times} ({X} ^ {6} + {Y} ^ {6})

The Dirichlet and Neumann conditions and the source term are obtained by the method of manufactured solutions [:ref:`bib1 <bib1>`].


Reference quantities and results
-----------------------------------

The value of the difference between analytical solutions and calculated on the mesh: :math:`\mathrm{\sum }^{\text{noeuds}n}\mathrm{\mid }{T}_{\text{n}}^{\text{calc}}\mathrm{-}{T}_{\text{n}}^{\text{ref}}\mathrm{\mid }`.

In the case of models that perform a convergence analysis with the fineness of the mesh, the speed of convergence with the fineness of the mesh from the solution calculated to the analytical solution in standard :math:`{L}_{2}`, i.e. the greatest real :math:`\alpha >0` such as :math:`{\mathrm{\parallel }{T}^{\text{calc}}\mathrm{-}{T}^{\text{ref}}\mathrm{\parallel }}_{\mathrm{0,}\Omega }<C\mathrm{\times }{h}^{\alpha }` where where :math:`C` is independent of :math:`h`.


Uncertainties about the solution
----------------------------

None


Bibliographical references
---------------------------


1. Document U2.08.08, Using the Manufactured Solutions Method for Software Validation, Code_Aster U2 Documentation