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The stimulus for the present work is the growing need for more
accurate numerical methods. The rapid advances in computer
technology have not provided the resources for computations which
make use of methods with low accuracy. The computational speed of
computers is continually increasing, while memory still remains a
problem when one handles large arrays. More accurate numerical
methods allow us to reduce the overall computation time by of
magnitude. several orders The problem of finding the most efficient
methods for the numerical solution of equations, under the
assumption of fixed array size, is therefore of paramount
importance. Advances in the applied sciences, such as aerodynamics,
hydrodynamics, particle transport, and scattering, have increased
the demands placed on numerical mathematics. New mathematical
models, describing various physical phenomena in greater detail
than ever before, create new demands on applied mathematics, and
have acted as a major impetus to the development of computer
science. For example, when investigating the stability of a fluid
flowing around an object one needs to solve the low viscosity form
of certain hydrodynamic equations describing the fluid flow. The
usual numerical methods for doing so require the introduction of a
"computational viscosity," which usually exceeds the physical
value; the results obtained thus present a distorted picture of the
phenomena under study. A similar situation arises in the study of
behavior of the oceans, assuming weak turbulence. Many additional
examples of this type can be given.
Multigrid Methods for Finite Elements combines two rapidly
developing fields: finite element methods, and multigrid
algorithms. At the theoretical level, Shaidurov justifies the rate
of convergence of various multigrid algorithms for self-adjoint and
non-self-adjoint problems, positive definite and indefinite
problems, and singular and spectral problems. At the practical
level these statements are carried over to detailed, concrete
problems, including economical constructions of triangulations and
effective work with curvilinear boundaries, quasilinear equations
and systems. Great attention is given to mixed formulations of
finite element methods, which allow the simplification of the
approximation of the biharmonic equation, the steady-state Stokes,
and Navier--Stokes problems.
Multigrid Methods for Finite Elements combines two rapidly
developing fields: finite element methods, and multigrid
algorithms. At the theoretical level, Shaidurov justifies the rate
of convergence of various multigrid algorithms for self-adjoint and
non-self-adjoint problems, positive definite and indefinite
problems, and singular and spectral problems. At the practical
level these statements are carried over to detailed, concrete
problems, including economical constructions of triangulations and
effective work with curvilinear boundaries, quasilinear equations
and systems. Great attention is given to mixed formulations of
finite element methods, which allow the simplification of the
approximation of the biharmonic equation, the steady-state Stokes,
and Navier--Stokes problems.
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