| TAZ-TFM-2017-915 |
Elementos finitos para problemas de punto silla
Pé de la Riva, Álvaro
Gaspar Lorenz, Francisco José (dir.) ; Rodrigo Cardiel, Carmen (dir.)
Universidad de Zaragoza,
CIEN,
2017
Matemática Aplicada department, Matemática Aplicada area
Máster Universitario en Modelización e Investigación Matemática, Estadística y Computación
Tipo de Trabajo Académico: Trabajo Fin de Master
Notas: 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.
Notas: 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.
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