82 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
Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation
In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann-Oden and for the symmetric DG method, we give a detailed analysis of the convergence for cell- and point-wise block-relaxation strategies. We show that, for a suitably constructed two-dimensional polynomial basis, point-wise block partitioning gives much better results than the classical cell-wise partitioning. Independent of the mesh size, for Poisson's equation, simple MG cycles, with block Gauss Seidel and symmetric block Gauss Seidel smoothing, yield a convergence rate of 0.4 - 0.6 per iteration sweep for both DG-methods studied
Non-invasive multigrid for semi-structured grids
Multigrid solvers for hierarchical hybrid grids (HHG) have been proposed to
promote the efficient utilization of high performance computer architectures.
These HHG meshes are constructed by uniformly refining a relatively coarse
fully unstructured mesh. While HHG meshes provide some flexibility for
unstructured applications, most multigrid calculations can be accomplished
using efficient structured grid ideas and kernels. This paper focuses on
generalizing the HHG idea so that it is applicable to a broader community of
computational scientists, and so that it is easier for existing applications to
leverage structured multigrid components. Specifically, we adapt the structured
multigrid methodology to significantly more complex semi-structured meshes.
Further, we illustrate how mature applications might adopt a semi-structured
solver in a relatively non-invasive fashion. To do this, we propose a formal
mathematical framework for describing the semi-structured solver. This
formalism allows us to precisely define the associated multigrid method and to
show its relationship to a more traditional multigrid solver. Additionally, the
mathematical framework clarifies the associated software design and
implementation. Numerical experiments highlight the relationship of the new
solver with classical multigrid. We also demonstrate the generality and
potential performance gains associated with this type of semi-structured
multigrid
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