49 research outputs found
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
Block preconditioning for fault/fracture mechanics saddle-point problems
The efficient simulation of fault and fracture mechanics is a key issue in several applications and is attracting a growing
interest by the scientific community. Using a formulation based on Lagrange multipliers, the Jacobian matrix
resulting from the Finite Element discretization of the governing equations has a non-symmetric generalized saddlepoint
structure. In this work, we propose a family of block preconditioners to accelerate the convergence of Krylov
methods for such problems. We critically review possible advantages and difficulties of using various Schur complement
approximations, based on both physical and algebraic considerations. The proposed approaches are tested
in a number of real-world applications, showing their robustness and efficiency also in large-size and ill-conditioned
problems
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
Enhanced multiscale restriction-smoothed basis (MsRSB) preconditioning with applications to porous media flow and geomechanics
A novel method to enable application of the Multiscale Restricted Smoothed
Basis (MsRSB) method to non M-matrices is presented. The original MsRSB method
is enhanced with a filtering strategy enforcing M-matrix properties to enable
the robust application of MsRSB as a preconditioner. Through applications to
porous media flow and linear elastic geomechanics, the method is proven to be
effective for scalar and vector problems with multipoint finite volume (FV) and
finite element (FE) discretization schemes, respectively. Realistic complex
(un)structured two- and three-dimensional test cases are considered to
illustrate the method's performance
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