11th GAMM-Workshop on

Multigrid and Hierarchic Solution Techniques


  A. Almendral  
  M. Bader  
  R. Bank  
  M. Bebendorf  
  S. Beuchler  
  D. Braess  
  C. Douglas  
  L. Grasedyck  
  B. Khoromskij  
  R. Kornhuber  
  B. Krukier  
  U. Langer  
  C. Oosterlee  
  G. Pöplau  
  A. Reusken  
  J. Schöberl  
  M.A. Schweitzer  
  S. Serra Capizzano  
  B. Seynaeve  
  D. Smits  
  O. Steinbach  
  R. Stevenson  
  M. Wabro  
  R. Wienands  
  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 point-wise smoothing method based on distributive iteration. In distributive smoothing the original system of equations is transformed by post-conditioning in order to achieve favorable properties, such as a decoupling of the equations and/or possibilities for point-wise 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 h-ellipticity 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 equation-wise 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 straight-forward 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 three-grid analysis [7]. Numerical test calculations validate the theoretical predictions.

  1. M.A. Biot, General theory of three dimensional consolidation, J. Appl. Physics, 12 (1941), pp. 155-16 4.
  2. A. Brandt, Multigrid techniques: 1984 guide with applications to fluid dynamics, GMD-Studie Nr. 85, Sankt Augustin, Germany, 1984.
  3. A. Brandt and N. Dinar, Multigrid solutions to elliptic flow problems, in: S. Parter, ed., Numerical methods for partial differential equations, Academic Press, New York, 1979, pp. 53-147.
  4. F.J. Gaspar, F.J. Lisbona, and P.N. Vabishchevich, A finite difference analysis of Biot's consolidation model, Appl. Num. Math. 44 (2003), pp. 487-506.
  5. V.C. Mow and W.M. Lai, Recent developments in synovial joint biomechanics, SIAM Review, 22 (1980), pp. 275-317.
  6. U. Trottenberg, C.W. Oosterlee, and A. Schller}, Multigrid, Academic Press, New York, 2001.
  7. R. Wienands and C.W. Oosterlee, On three-grid Fourier analysis for multigrid, SIAM J. Sci. Comput., 23 (2001), pp. 651-671.
  8. G. Wittum, Multi-grid methods for Stokes and Navier-Stokes equations with transforming smoothers: algorithms and numerical results, Num. Math., 54 (1989), pp. 543-563.

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