305 research outputs found
Multigrid Methods for Saddle Point Problems: Darcy Systems
We design and analyze multigrid methods for the saddle point problems
resulting from Raviart-Thomas-N\'ed\'elec mixed finite element methods (of
order at least 1) for the Darcy system in porous media flow. Uniform
convergence of the -cycle algorithm in a nonstandard energy norm is
established. Extensions to general second order elliptic problems are also
addressed
Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems
A V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ)
discretization of the Kirchhoff plate bending problems is developed in this
paper. It is shown that the contraction number of the V-cycle multigrid HHJ
mixed method is bounded away from one uniformly with respect to the mesh size.
The uniform convergence is achieved for the V-cycle multigrid method with only
one smoothing step and without full elliptic regularity. The key is a stable
decomposition of the kernel space which is derived from an exact sequence of
the HHJ mixed method, and the strengthened Cauchy Schwarz inequality. Some
numerical experiments are provided to confirm the proposed V-cycle multigrid
method. The exact sequences of the HHJ mixed method and the corresponding
commutative diagram is of some interest independent of the current context.Comment: 23 page
Solution of nonlinear Stokes equations discretized by high-order finite elements on nonconforming and anisotropic meshes, with application to ice sheet dynamics
Motivated by the need for efficient and accurate simulation of the dynamics
of the polar ice sheets, we design high-order finite element discretizations
and scalable solvers for the solution of nonlinear incompressible Stokes
equations. We focus on power-law, shear thinning rheologies used in modeling
ice dynamics and other geophysical flows. We use nonconforming hexahedral
meshes and the conforming inf-sup stable finite element velocity-pressure
pairings or . To solve the nonlinear equations, we
propose a Newton-Krylov method with a block upper triangular preconditioner for
the linearized Stokes systems. The diagonal blocks of this preconditioner are
sparse approximations of the (1,1)-block and of its Schur complement. The
(1,1)-block is approximated using linear finite elements based on the nodes of
the high-order discretization, and the application of its inverse is
approximated using algebraic multigrid with an incomplete factorization
smoother. This preconditioner is designed to be efficient on anisotropic
meshes, which are necessary to match the high aspect ratio domains typical for
ice sheets. We develop and make available extensions to two libraries---a
hybrid meshing scheme for the p4est parallel AMR library, and a modified
smoothed aggregation scheme for PETSc---to improve their support for solving
PDEs in high aspect ratio domains. In a numerical study, we find that our
solver yields fast convergence that is independent of the element aspect ratio,
the occurrence of nonconforming interfaces, and of mesh refinement, and that
depends only weakly on the polynomial finite element order. We simulate the ice
flow in a realistic description of the Antarctic ice sheet derived from field
data, and study the parallel scalability of our solver for problems with up to
383M unknowns.Comment: 31 page
A two-level algorithm for the weak Galerkin discretization of diffusion problems
This paper analyzes a two-level algorithm for the weak Galerkin (WG) finite
element methods based on local Raviart-Thomas (RT) and Brezzi-Douglas-Marini
(BDM) mixed elements for two- and three-dimensional diffusion problems with
Dirichlet condition. We first show the condition numbers of the stiffness
matrices arising from the WG methods are of . We use an extended
version of the Xu-Zikatanov (XZ) identity to derive the convergence of the
algorithm without any regularity assumption. Finally we provide some numerical
results
Unified geometric multigrid algorithm for hybridized high-order finite element methods
We consider a standard elliptic partial differential equation and propose a
geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for
hybridized high-order finite element methods. The proposed unified approach is
applicable to any locally conservative hybridized finite element method
including multinumerics with different hybridized methods in different parts of
the domain. For these methods, the linear system involves only the unknowns
residing on the mesh skeleton, and constructing intergrid transfer operators is
therefore not trivial. The key to our geometric multigrid algorithm is the
physics-based energy-preserving intergrid transfer operators which depend only
on the fine scale DtN maps. Thanks to these operators, we completely avoid
upscaling of parameters and no information regarding subgrid physics is
explicitly required on coarse meshes. Moreover, our algorithm is
agglomeration-based and can straightforwardly handle unstructured meshes. We
perform extensive numerical studies with hybridized mixed methods, hybridized
discontinuous Galerkin method, weak Galerkin method, and a hybridized version
of interior penalty discontinuous Galerkin methods on a range of elliptic
problems including subsurface flow through highly heterogeneous porous media.
We compare the performance of different smoothers and analyze the effect of
stabilization parameters on the scalability of the multigrid algorithm
Fast multilevel solvers for a class of discrete fourth order parabolic problems
In this paper, we study fast iterative solvers for the solution of fourth
order parabolic equations discretized by mixed finite element methods. We
propose to use consistent mass matrix in the discretization and use lumped mass
matrix to construct efficient preconditioners. We provide eigenvalue analysis
for the preconditioned system and estimate the convergence rate of the
preconditioned GMRes method. Furthermore, we show that these preconditioners
only need to be solved inexactly by optimal multigrid algorithms. Our numerical
examples indicate that the proposed preconditioners are very efficient and
robust with respect to both discretization parameters and diffusion
coefficients. We also investigate the performance of multigrid algorithms with
either collective smoothers or distributive smoothers when solving the
preconditioner systems.Comment: 27 page
Fast Auxiliary Space Preconditioner for Linear Elasticity in Mixed Form
A block diagonal preconditioner with the minimal residual method and a block
triangular preconditioner with the generalized minimal residual method are
developed for Hu-Zhang mixed finite element methods of linear elasticity. They
are based on a new stability result of the saddle point system in
mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the
mass matrix which is easy to invert while the displacement it is spectral
equivalent to Schur complement. A fast auxiliary space preconditioner based on
the conforming linear element of the linear elasticity problem is then
designed for solving the Schur complement. For both diagonal and triangular
preconditioners, it is proved that the conditioning numbers of the
preconditioned systems are bounded above by a constant independent of both the
crucial Lam\'e constant and the mesh-size. Numerical examples are presented to
support theoretical results. As byproducts, a new stabilized low order mixed
finite element method is proposed and analyzed and superconvergence results of
Hu-Zhang element are obtained.Comment: 25 page
V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes
In this paper we analyse the convergence properties of V-cycle multigrid
algorithms for the numerical solution of the linear system of equations arising
from discontinuous Galerkin discretization of second-order elliptic partial
differential equations on polytopal meshes. Here, the sequence of spaces that
stands at the basis of the multigrid scheme is possibly non nested and is
obtained based on employing agglomeration with possible edge/face coarsening.
We prove that the method converges uniformly with respect to the granularity of
the grid and the polynomial approximation degree p, provided that the number of
smoothing steps, which depends on p, is chosen sufficiently large.Comment: 26 pages, 23 figures, submitted to Journal of Scientific Computin
A new extrapolation cascadic multigrid method for 3D elliptic boundary value problems on rectangular domains
In this paper, we develop a new extrapolation cascadic multigrid
(ECMG) method, which makes it possible to solve 3D elliptic boundary
value problems on rectangular domains of over 100 million unknowns on a desktop
computer in minutes. First, by combining Richardson extrapolation and
tri-quadratic Serendipity interpolation techniques, we introduce a new
extrapolation formula to provide a good initial guess for the iterative
solution on the next finer grid, which is a third order approximation to the
finite element (FE) solution. And the resulting large sparse linear system from
the FE discretization is then solved by the Jacobi-preconditioned Conjugate
Gradient (JCG) method. Additionally, instead of performing a fixed number of
iterations as cascadic multigrid (CMG) methods, a relative residual stopping
criterion is used in iterative solvers, which enables us to obtain conveniently
the numerical solution with the desired accuracy. Moreover, a simple Richardson
extrapolation is used to cheaply get a fourth order approximate solution on the
entire fine grid. Test results are reported to show that ECMG has much
better efficiency compared to the classical MG methods. Since the initial guess
for the iterative solution is a quite good approximation to the FE solution,
numerical results show that only few number of iterations are required on the
finest grid for ECMG with an appropriate tolerance of the relative
residual to achieve full second order accuracy, which is particularly important
when solving large systems of equations and can greatly reduce the
computational cost. It should be pointed out that when the tolerance becomes
smaller, ECMG still needs only few iterations to obtain fourth order
extrapolated solution on each grid, except on the finest grid. Finally, we
present the reason why our ECMG algorithms are so highly efficient for solving
such problems.Comment: 20 pages, 4 figures, 10 tables; abbreviated abstrac
A Multilevel Approach for Trace System in HDG Discretizations
We propose a multilevel approach for trace systems resulting from hybridized
discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested
dissection, domain decomposition, and high-order characteristic of HDG
discretizations. Specifically, we first create a coarse solver by eliminating
and/or limiting the front growth in nested dissection. This is accomplished by
projecting the trace data into a sequence of same or high-order polynomials on
a set of increasingly coarser edges/faces. We then combine the coarse
solver with a block-Jacobi fine scale solver to form a two-level
solver/preconditioner. Numerical experiments indicate that the performance of
the resulting two-level solver/preconditioner depends only on the smoothness of
the solution and is independent of the nature of the PDE under consideration.
While the proposed algorithms are developed within the HDG framework, they are
applicable to other hybrid(ized) high-order finite element methods. Moreover,
we show that our multilevel algorithms can be interpreted as a multigrid method
with specific intergrid transfer and smoothing operators. With several
numerical examples from Poisson, pure transport, and convection-diffusion
equations we demonstrate the robustness and scalability of the algorithms
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