Jean Deteix - Numerical Methods for Mixed Finite Element Problems: Applications to Incompressible Materials and Contact Problems
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- Book:Numerical Methods for Mixed Finite Element Problems: Applications to Incompressible Materials and Contact Problems
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Starting by recalling the basic theory of mixed finite element methods, the book goes on to discuss the augmented Lagrangian method and gives a summary of the standard iterative methods, describing their usage for mixed methods. Here, preconditioners are built from an approximate factorisation of the mixed system.
A first set of applications is considered for incompressible elasticity problems and flow problems, including non-linear models.
An account of the mixed formulation for Dirichlets boundary conditions is then given before turning to contact problems, where contact between incompressible bodies leads to problems with two constraints.
This book is aimed at graduate students and researchers in the field of numerical methods and scientific computing.
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This series reports on new developments in all areas of mathematics and their applications - quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes:
1. Research monographs
2. Lectures on a new field or presentations of a new angle in a classical field
3. Summer schools and intensive courses on topics of current research.
Texts which are out of print but still in demand may also be considered if they fall within these categories. The timeliness of a manuscript is sometimes more important than its form, which may be preliminary or tentative.
Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews, and zbMATH.
This Springer imprint is published by the registered company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Mixed Finite Element Methods are often discarded because they lead to indefinite systems which are more difficult to solve than the nice positive definite problems of standard methods. Indeed, solving indefinite systems is a challenge : direct methods []. As an example, consider the classical conjugate gradient method. Applied to a symmetric indefinite problem it will generate a diverging sequence. As the conjugate gradient method is (in exact arithmetic) a direct method, it will yield the exact solution if the problem is small enough to avoid loosing orthogonality. Applying a minimum residual method to the same problem will in most cases yield stagnation.
These two classical methods are the simplest in a list which grows constantly. This monograph does not intend to introduce new iteration methods. We shall rely mostly on existing packages, mostly Petsc from Argonne Laboratory [].
The key to obtain convergence is preconditioning. For general problem, a vast number of preconditioners are available [).
We want to show in the present work that efficient iterative methods can be developed for this class of problems and that they make possible the solution of large problems with both accuracy and efficiency. We shall also insist on the fact that these methods should be entirely automatic and free of user dependent parameters.
We also want to make clear that our numerical results should be taken as examples and that we do not claim that they are optimal. Our hope is that they could be a starting point for further research.
Here is therefore our plan.
Chapter 1 rapidly recalls the classical theory of mixed problems, including Augmented Lagrangian methods and their matricial form.
Chapter presents some classical iterative methods and describes the preconditioner which will be central to our development. We come back to augmented Lagrangian and a mixed form for penalty methods.
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