Abstract
We discuss the construction of robust preconditioners for finite element approximations of Biot’s consolidation model in poroelasticity. More precisely, we study finite element methods based on generalizations of the Hellinger–Reissner principle of linear elasticity, where the stress tensor is one of the unknowns. The Biot model has a number of applications in science, medicine, and engineering. A challenge in many of these applications is that the model parameters range over several orders of magnitude. Therefore, discretization procedures which are well behaved with respect to such variations are needed. The focus of the present paper will be on the construction of preconditioners, such that the preconditioned discrete systems are well-conditioned with respect to variations of the model parameters as well as refinements of the discretization. As a byproduct, we also obtain preconditioners for linear elasticity that are robust in the incompressible limit.
Funding statement: The research leading to these results has received funding the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 339643. The work of Kent-Andre Mardal has also been supported by the Research Council of Norway through grant no. 179578/F50.
A Appendix: Proofs of (3.3) and (3.5)
Here we provide proofs of inequalities (3.3) and (3.5).
Proof of (3.3).
Fix
and the fact that
for some constant C independent of τ.
To prove this, we use a well-known result for the right inverse of the divergence operator: There exists
(A.1)
with
Since
where the first term of the final form is a result of integration by parts. Next, we may use the Cauchy–Schwarz inequality, which results in
and so the result follows after dividing by
When
Proof of (3.5).
Fix any
we have that
Since
with
B Appendix: Right Inverse of Divergence Operator
A result for the right inverse of the divergence operator, as expressed by (A.1), is closely related to the inf-sup condition for the Stokes problem, and therefore well-known. However, we are not aware of a proper reference for the case when
Lemma B.1.
Assume that
Proof.
Take any
Consider then the problem of finding
By the Lax–Milgram lemma (cf. e.g. [8, Theorem 4.1.6]), problem (B.1) has a unique solution ζ and
Therefore, if we set
we have
and
where the constant
which completes the proof. ∎
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