@article{PédelaRiva:63652,
author = "Pé de la Riva, Álvaro and Gaspar Lorenz, Francisco José
and Rodrigo Cardiel, Carmen",
title = "{Elementos finitos para problemas de punto silla}",
year = "2017",
note = "In the context of PDEs approximation, saddle point
problems take an important place. This type of problems
appear when the weak formulation of a PDE system shows a
special structure. Moreover, these problems can be
ill-posed: There might not be unique solvability for every
right hand side. They can be formulated in Hilbert spaces
as a model variational problem, that we describe in Chapter
2. At this level, their existence and uniqueness of
solution are not always ensured. That is the case when the
so-called inf-sup conditions do not hold. However, when
these conditions are satisfied, one is even able to bound
the solution using constants related to them. This topic
will be developed in Chapter 3, where some estimates for
the solution of a saddle point problem are given. Besides,
Chapter 4 is devoted to the approximation of saddle point
problems in finite dimensional spaces. One important issue
is the fact that the fulfillment of the inf-sup conditions
on Hilbert spaces does not imply the fulfillment of their
discrete version on finite dimensional subspaces. Due to
this, one has to be careful when choosing those subspaces.
Furthermore, this choice may yield better or worse
estimates depending on the size of the discrete inf-sup
constants. The discretisation of these problems using a
numerical method, like the finite element method, yields a
linear system whose matrix is called saddle point matrix.
Again, we find particular properties on them. As we will
see along Chapter 5, although one is interested on
invertibility of these matrices, they are indefinite. Thus,
one has to make a further study about solvability
conditions. Among the PDEs whose variational formulation
yields a saddle point problem, we find the Stokes
equations. In order to discretize the Stokes equations, it
is usual to apply mixed finite element methods. This fact
is due to the non fulfillment of the discrete inf-sup
conditions if one applies the same finite element spaces
for both variables. In addition, not every pair of finite
element spaces guarantees the stability of the method. For
instance, some troubles may take place like the spurious
pressure modes and the locking phenomenon. This will be
studied in Chapter 6, where we will comment on the
stability of several finite element pairs for the Stokes
equations. Finally, the choice of a good finite element
pair implies satisfactory numerical results and expected
convergence rates. That is the case of the Minielement and
the Taylor-Hood finite element method for the Stokes
equations. In order to prove this statement, we show the
obtained numerical results in Chapter 7. Furthermore, the
interested reader can find the implementation of these
mixed finite element methods in the appendices.",
}