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Non-standard finite element methods, in particular mixed methods,
are central to many applications. In this text the authors, Boffi,
Brezzi and Fortin present a general framework, starting with a
finite dimensional presentation, then moving on to formulation in
Hilbert spaces and finally considering approximations, including
stabilized methods and eigenvalue problems. This book also provides
an introduction to standard finite element approximations, followed
by the construction of elements for the approximation of mixed
formulations in H(div) and H(curl). The general theory is applied
to some classical examples: Dirichlet's problem, Stokes' problem,
plate problems, elasticity and electromagnetism.
Research on non-standard finite element methods is evolving rapidly
and in this text Brezzi and Fortin give a general framework in
which the development is taking place. The presentation is built
around a few classic examples: Dirichlet's problem, Stokes problem,
Linear elasticity. The authors provide with this publication an
analysis of the methods in order to understand their properties as
thoroughly as possible.
Since the early 70's, mixed finite elements have been the object
of a wide and deep study by the mathematical and engineering
communities. The fundamental role of this method for many
application fields has been worldwide recognized and its use has
been introduced in several commercial codes. An important feature
of mixed finite elements is the interplay between theory and
application. Discretization spaces for mixed schemes require
suitable compatibilities, so that simple minded approximations
generally do not work and the design of appropriate stabilizations
gives rise to challenging mathematical problems.
This volume collects the lecture notes of a C.I.M.E. course held
in Summer 2006, when some of the most world recognized experts in
the field reviewed the rigorous setting of mixed finite elements
and revisited it after more than 30 years of practice.
Applications, in this volume, range from traditional ones, like
fluid-dynamics or elasticity, to more recent and active fields,
like electromagnetism.
This book focuses on iterative solvers and preconditioners for
mixed finite element methods. It provides an overview of some of
the state-of-the-art solvers for discrete systems with constraints
such as those which arise from mixed formulations. 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 Dirichlet's 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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