

Roman Wienands:Efficient multigrd solution for the system of poroelasticity equations
Poroelasticity has a wide range of applications in biology, filtration, and soil sciences.
The mathematical model for a general situation was first proposed and analyzed by Biot [1],
studying the consolidation of soils. Poroelastic models are used nowadays to study problems
in geomechanics, hydrogeology and petrol engineering. The equations have recently been applied
in biomechanics to the study of soft tissue compression [5], to model the deformation and
permeability of biological tissues, such as cartilage, skin, lungs, arterial or myocardial tissues.
In this talk, we present an efficient multigrid method for the system of poroelasticity equations
in two and three dimensions. In particular, we introduce a pointwise smoothing method based
on distributive iteration. In distributive smoothing the original system of equations is transformed
by postconditioning in order to achieve favorable properties, such as a decoupling of the
equations and/or possibilities for pointwise smoothing [2,3,8]. A specialty lies in the
discretization approach employed. We adopt a staggered grid for the poroelasticity equations,
whereas the usual way to discretize the equations is by means of finite elements. Standard finite
elements (or finite differences), however, do not lead to stable solutions without additional
stabilization. The staggered grid discretization, as proposed in [4], leads to a natural stable
discretization for the poroelasticity system.
The multigrid method is developed with analysis of increasing complexity on the basis of Fourier
analysis [2,6]. After the analysis of the determinant of the system of equations, the hellipticity
concept is discussed, which is fundamental for the existence of point smoothers. The smoother
is evaluated and relaxation parameters are obtained with Fourier smoothing analysis. An
equationwise decoupled smoother with as principal operators simple Laplacian and biharmonic operators
is obtained, whereas the diagonal blocks in the system of poroelasticity equations may contain
anisotropies. The coarse grid correction is based on straightforward geometric transfer operators
and coarse grid analogs of the fine grid discretiz ation. The resulting multigrid method is
evaluated with the help of the Fourier threegrid analysis [7]. Numerical test calculations
validate the theoretical predictions.
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methods for partial differential equations, Academic Press, New York, 1979, pp. 53147.
 F.J. Gaspar, F.J. Lisbona, and P.N. Vabishchevich, A finite difference analysis of Biot's
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 V.C. Mow and W.M. Lai, Recent developments in synovial joint biomechanics, SIAM Review, 22 (1980),
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 U. Trottenberg, C.W. Oosterlee, and A. Schller}, Multigrid, Academic Press, New York, 2001.
 R. Wienands and C.W. Oosterlee, On threegrid Fourier
analysis for multigrid, SIAM J. Sci. Comput., 23 (2001), pp. 651671.
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