38 research outputs found
Robust preconditioners for a new stabilized discretization of the poroelastic equations
In this paper, we present block preconditioners for a stabilized
discretization of the poroelastic equations developed in [45]. The
discretization is proved to be well-posed with respect to the physical and
discretization parameters, and thus provides a framework to develop
preconditioners that are robust with respect to such parameters as well. We
construct both norm-equivalent (diagonal) and field-of-value-equivalent
(triangular) preconditioners for both the stabilized discretization and a
perturbation of the stabilized discretization that leads to a smaller overall
problem after static condensation. Numerical tests for both two- and
three-dimensional problems confirm the robustness of the block preconditioners
with respect to the physical and discretization parameters
Efficient solvers for hybridized three-field mixed finite element coupled poromechanics
We consider a mixed hybrid finite element formulation for coupled
poromechanics. A stabilization strategy based on a macro-element approach is
advanced to eliminate the spurious pressure modes appearing in
undrained/incompressible conditions. The efficient solution of the stabilized
mixed hybrid block system is addressed by developing a class of block
triangular preconditioners based on a Schur-complement approximation strategy.
Robustness, computational efficiency and scalability of the proposed approach
are theoretically discussed and tested using challenging benchmark problems on
massively parallel architectures
Hybrid mimetic finite-difference and virtual element formulation for coupled poromechanics
We present a hybrid mimetic finite-difference and virtual element formulation
for coupled single-phase poromechanics on unstructured meshes. The key
advantage of the scheme is that it is convergent on complex meshes containing
highly distorted cells with arbitrary shapes. We use a local pressure-jump
stabilization method based on unstructured macro-elements to prevent the
development of spurious pressure modes in incompressible problems approaching
undrained conditions. A scalable linear solution strategy is obtained using a
block-triangular preconditioner designed specifically for the saddle-point
systems arising from the proposed discretization. The accuracy and efficiency
of our approach are demonstrated numerically on two-dimensional benchmark
problems.Comment: 25 pages, 17 figure
Parameter-robust discretization and preconditioning of Biot's consolidation model
Biot's consolidation model in poroelasticity has a number of applications in
science, medicine, and engineering. The model depends on various parameters,
and in practical applications these parameters ranges over several orders of
magnitude. A current challenge is to design discretization techniques and
solution algorithms that are well behaved with respect to these variations. The
purpose of this paper is to study finite element discretizations of this model
and construct block diagonal preconditioners for the discrete Biot systems. The
approach taken here is to consider the stability of the problem in non-standard
or weighted Hilbert spaces and employ the operator preconditioning approach. We
derive preconditioners that are robust with respect to both the variations of
the parameters and the mesh refinement. The parameters of interest are small
time-step sizes, large bulk and shear moduli, and small hydraulic conductivity.Comment: 24 page
Parameter robust preconditioning by congruence for multiple-network poroelasticity
The mechanical behaviour of a poroelastic medium permeated by multiple
interacting fluid networks can be described by a system of time-dependent
partial differential equations known as the multiple-network poroelasticity
(MPET) equations or multi-porosity/multi-permeability systems. These equations
generalize Biot's equations, which describe the mechanics of the one-network
case. The efficient numerical solution of the MPET equations is challenging, in
part due to the complexity of the system and in part due to the presence of
interacting parameter regimes. In this paper, we present a new strategy for
efficiently and robustly solving the MPET equations numerically. In particular,
we introduce a new approach to formulating finite element methods and
associated preconditioners for the MPET equations. The approach is based on
designing transformations of variables that simultaneously diagonalize (by
congruence) the equations' key operators and subsequently constructing
parameter-robust block-diagonal preconditioners for the transformed system. Our
methodology is supported by theoretical considerations as well as by numerical
results