NUMERICAL METHODS
FOR THE BIOT MODEL IN POROELASTICITY

A Dissertation
presented to
the Faculty of the Department of Mathematics
University of Houston

by

Konstantin Lipnikov

May 2002


Title and Abstract

1. Introduction ............................................. 1
  1.1. Review of discretization methods ....................  2
  1.2. Review of solution methods ..........................  5
  1.3. Thesis outline ......................................  7 
2. The Biot model in poroelasticity ........................ 10
  2.1. Mathematical model .................................. 10
  2.2. Finite element discretization ....................... 13
  2.3. Mimetic finite difference discretization ............ 19
3. Numerical methods for coupled problemy .................. 27
  3.1. Block diagonal preconditioners ...................... 29
  3.2. Preconditioners for the elastic block ............... 31
  3.3. Preconditioners for the diffusion block ............. 32
    3.3.1. Preconditioner for matrix A_{ff} ................ 32
    3.3.2. Preconditioner for Schur complement S_p ......... 33
  3.4. Numerical experiments ............................... 43
4. Numerical methods for hybridized problem ................ 47
  4.2. Separable preconditioner for A_{\lambda \lambda} .... 52
    4.2.1. Fast direct method for 2D problems .............. 53
    4.2.2. Applications of 2D algorithm .................... 60
    4.2.3. Fast direct method for 3D problems .............. 66
    4.2.4. Applications of 3D algorithm .................... 69
  4.3. Multilevel preconditioner for A_{\lambda \lambda} ... 72
    4.3.1. Abstract form of multilevel algorithm ........... 72
    4.3.2. Algorithm implementation ........................ 79
    4.3.3. Numerical experiments ........................... 83
5. On accuracy of mimetic discretizations .................. 85
  5.1. Error estimates for the mixed FE method ............. 86
  5.2. Error estimates for the SO method ................... 95
  5.3. Numerical experiments .............................. 104
Bibliography .............................................. 108