581 research outputs found
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems
In this paper we develop a family of preconditioners for the linear algebraic
systems arising from the arbitrary Lagrangian-Eulerian discretization of some
fluid-structure interaction models. After the time discretization, we formulate
the fluid-structure interaction equations as saddle point problems and prove
the uniform well-posedness. Then we discretize the space dimension by finite
element methods and prove their uniform well-posedness by two different
approaches under appropriate assumptions. The uniform well-posedness makes it
possible to design robust preconditioners for the discretized fluid-structure
interaction systems. Numerical examples are presented to show the robustness
and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2.
Rerun all the numerical tests 3. changed title, abstract and corrected lots
of typos and inconsistencies 4. added reference
Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Saddle Point Problems
Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes, Navier-Stokes, and mixed linear elasticity ty e are presented. For the first time, coarse spaces obtained from the GDSW (Generalized Dryja-Smith-Widlund) approach are used in such a setting. Numerical results of our parallel implementation are presented for several model problems. In particular, cases are considered where the problem cannot or should not b e reduced using local static condensation, e.g., Stokes, Navier-Stokes or mixed elasticity problems with continuous pressure spaces. In the new monolithic preconditioners, the local overlapping problems and the coarse problem are saddle point problems with the same structure as the original problem. Our parallel implementation of these preconditioners is based on the FROSch (Fast and Robust Overlapping Schwarz) library, which is part of the Trilinos package ShyLU. The implementation is algebraic in the sense that the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Parallel scalability results for several thousand cores for Stokes, Navier-Stokes, and mixed linear elasticity model problems are reported. Each of the local problems is solved using a direct solver in serial mo de, whereas the coarse problem is solved using a direct solver in serial or MPI-parallel mode or using an MPI-parallel iterative Krylov solve
HAZniCS -- Software Components for Multiphysics Problems
We introduce the software toolbox HAZniCS for solving interface-coupled
multiphysics problems. HAZniCS is a suite of modules that combines the
well-known FEniCS framework for finite element discretization with solver and
graph library HAZmath. The focus of the paper is on the design and
implementation of a pool of robust and efficient solver algorithms which tackle
issues related to the complex interfacial coupling of the physical problems
often encountered in applications in brain biomechanics. The robustness and
efficiency of the numerical algorithms and methods is shown in several
numerical examples, namely the Darcy-Stokes equations that model flow of
cerebrospinal fluid in the human brain and the mixed-dimensional model of
electrodiffusion in the brain tissue
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