Ill-conditioning of finite element poroelasticity equations
M. Ferronato, G. Gambolati, P. Teatini
Dept. Mathematical Methods and Models for Scientific
Applications, University of Padova, Padova, Italy
ABSTRACT
The solution to Biot's coupled consolidation theory is usually addressed by
the Finite Element (FE) method thus obtaining a system of first-order
differential equations which is integrated by the use of an appropriate
time marching scheme. For small values of the time step the resulting
linear system may be severely ill-conditioned and hence the solution
can prove quite difficult to achieve. Under such conditions efficient and
robust projection solvers based on Krylov's subspaces which are usually
recommended for non-symmetric large
size problems can exhibit a very slow convergence rate or even fail.
The present paper investigates the correlation between the
ill-conditioning of FE poroelasticity equations and the time integration step
Dt. An empirical relation is provided for a lower bound
Dtcrit of Dt below which ill-conditioning may suddenly
occur. The critical time step is larger for soft and low
permeable porous media discretized on coarser grids. A limiting value
for the rock stiffness is found such that for stiffer systems there is
no ill-conditioning irrespective of Dt however small, as is also
shown by several numerical examples.
Finally, the definition of a different Dtcrit as suggested by
other authors is reviewed and discussed.
