819 research outputs found
Poroelasticity problem: numerical difficulties and efficient multigrid solution
This work contains some of the more relevant results obtained by the author regarding the numerical solution of the Biot’s consolidation problem. The emphasis here is on the stable discretization and the highly efficient solution of the resulting algebraic system of equations, which is of saddle point type. On the one hand, a stabilized linear finite element scheme providing oscillation-free solutions for this model is proposed and theoretically analyzed. On the other hand, a monolithic multigrid method is considered for the solution of the resulting system of equations after discretization by using the stabilized scheme. Since this system is of saddle point type, special smoothers of “Vanka”-type have to be considered. This multigrid method is designed with the help of an special local Fourier analysis that takes into account the specific characteristics of the considered block-relaxations. Results from this analysis are presented and compared with those experimentally computed
A local Fourier analysis of additive Vanka relaxation for the Stokes equations
Multigrid methods are popular solution algorithms for many discretized PDEs,
either as standalone iterative solvers or as preconditioners, due to their high
efficiency. However, the choice and optimization of multigrid components such
as relaxation schemes and grid-transfer operators is crucial to the design of
optimally efficient algorithms. It is well--known that local Fourier analysis
(LFA) is a useful tool to predict and analyze the performance of these
components. In this paper, we develop a local Fourier analysis of monolithic
multigrid methods based on additive Vanka relaxation schemes for mixed
finite-element discretizations of the Stokes equations. The analysis offers
insight into the choice of "patches" for the Vanka relaxation, revealing that
smaller patches offer more effective convergence per floating point operation.
Parameters that minimize the two-grid convergence factor are proposed and
numerical experiments are presented to validate the LFA predictions.Comment: 30 pages, 12 figures. Add new sections: multiplicative Vanka results
and sensitivity of convergence factors to mesh distortio
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
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