Summary of results in 2D
============================


+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+
|                                                                                   |                                                                                   |**TRIA3**|**QUAD4**|**TRIA6**|**QUAD8**|**QUAD9**|
+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+
|                                                                                   |Exact                                                                              |3.44%    |2.48%    |1.17%    |0.697%   |0.695%   |
+ .. image:: images/Object_130.svg                                                  +                                                                                   +         +         +         +         +         +
|    :width: 25                                                                     |                                                                                   |         |         |         |         |         |
+    :height: 22                                                                    +                                                                                   +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+                                                                                   +                                                                                   +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+
|                                                                                   |                                                                                   |3.28%    |2.28%    |0.95%    |0.22%    |0.21%    |
+                                                                                   + .. image:: images/Object_131.svg                                                  +         +         +         +         +         +
|                                                                                   |    :width: 25                                                                     |         |         |         |         |         |
+                                                                                   +    :height: 22                                                                    +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+                                                                                   +                                                                                   +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+
|                                                                                   |                                                                                   |3.49%    |2.37%    |1.29%    |0.687%   |0.66%    |
+                                                                                   + .. image:: images/Object_132.svg                                                  +         +         +         +         +         +
|                                                                                   |    :width: 25                                                                     |         |         |         |         |         |
+                                                                                   +    :height: 22                                                                    +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+                                                                                   +                                                                                   +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+
|                                                                                   |                                                                                   |0.952    |0.919    |0.810    |0.321    |0.306    |
+ .. image:: images/Object_133.svg                                                  + .. image:: images/Object_134.svg                                                  +         +         +         +         +         +
|    :width: 25                                                                     |    :width: 30                                                                     |         |         |         |         |         |
+    :height: 22                                                                    +    :height: 15                                                                    +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+                                                                                   +                                                                                   +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+
|                                                                                   |                                                                                   |1.013    |0.958    |1.099    |0.985    |0.951    |
+                                                                                   + .. image:: images/Object_135.svg                                                  +         +         +         +         +         +
|                                                                                   |    :width: 25                                                                     |         |         |         |         |         |
+                                                                                   +    :height: 22                                                                    +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+                                                                                   +                                                                                   +         +         +         +         +         +
|                                                                                   |                                                                                   |         |         |         |         |         |
+-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+---------+---------+---------+---------+---------+


Node constraints are, within the set, best approximated with :math:`\mathrm{ZZ2}`, especially for order 2 items. If we make it tender

.. image:: images/Object_136.svg
    :width: 25
    :height: 22


.. _RefImage_Object_136.svg:

towards 0, the convergence rates with :math:`h` of

.. image:: images/Object_137.svg
    :width: 25
    :height: 22


.. _RefImage_Object_137.svg:

are greater by the :math:`\mathrm{ZZ2}` method for all element types than the :math:`\mathrm{ZZ1}` method (

.. image:: images/Object_138.svg
    :width: 25
    :height: 22


.. _RefImage_Object_138.svg:

is the smoothed stress).

The estimator :math:`\mathrm{ZZ1}` is not reliable for elements of order 2, the nodal constraints remain correct. In this particular case, it can be verified that

.. image:: images/Object_139.svg
    :width: 25
    :height: 22


.. _RefImage_Object_139.svg:

when

.. image:: images/Object_140.svg
    :width: 25
    :height: 22


.. _RefImage_Object_140.svg:

, which shows that continuous global smoothing is insufficient to estimate the error in the case of a solution without a singularity (case of this test).

On the other hand, :math:`\mathrm{ZZ2}` is reliable and asymptotically accurate (

.. image:: images/Object_141.svg
    :width: 25
    :height: 22


.. _RefImage_Object_141.svg:

when

.. image:: images/Object_142.svg
    :width: 25
    :height: 22


.. _RefImage_Object_142.svg:

).