485 research outputs found
Robust Preconditioners for Incompressible MHD Models
In this paper, we develop two classes of robust preconditioners for the
structure-preserving discretization of the incompressible magnetohydrodynamics
(MHD) system. By studying the well-posedness of the discrete system, we design
block preconditioners for them and carry out rigorous analysis on their
performance. We prove that such preconditioners are robust with respect to most
physical and discretization parameters. In our proof, we improve the existing
estimates of the block triangular preconditioners for saddle point problems by
removing the scaling parameters, which are usually difficult to choose in
practice. This new technique is not only applicable to the MHD system, but also
to other problems. Moreover, we prove that Krylov iterative methods with our
preconditioners preserve the divergence-free condition exactly, which
complements the structure-preserving discretization. Another feature is that we
can directly generalize this technique to other discretizations of the MHD
system. We also present preliminary numerical results to support the
theoretical results and demonstrate the robustness of the proposed
preconditioners
The auxiliary space preconditioner for the de Rham complex
We generalize the construction and analysis of auxiliary space
preconditioners to the n-dimensional finite element subcomplex of the de Rham
complex. These preconditioners are based on a generalization of a decomposition
of Sobolev space functions into a regular part and a potential. A discrete
version is easily established using the tools of finite element exterior
calculus. We then discuss the four-dimensional de Rham complex in detail. By
identifying forms in four dimensions (4D) with simple proxies, form operations
are written out in terms of familiar algebraic operations on matrices, vectors,
and scalars. This provides the basis for our implementation of the
preconditioners in 4D. Extensive numerical experiments illustrate their
performance, practical scalability, and parameter robustness, all in accordance
with the theory
A simple preconditioner for a discontinuous Galerkin method for the Stokes problem
In this paper we construct Discontinuous Galerkin approximations of the
Stokes problem where the velocity field is H(div)-conforming. This implies that
the velocity solution is divergence-free in the whole domain. This property can
be exploited to design a simple and effective preconditioner for the final
linear system.Comment: 27 pages, 4 figure
Auxiliary space preconditioning in H 0(curl; Ω)
We adapt the principle of auxiliary space preconditioning as presented in [J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing, 56 (1996), pp. 215-235.] to H (curl; ω)-elliptic variational problems discretized by means of edge elements. The focus is on theoretical analysis within the abstract framework of subspace correction. Employing a Helmholtz-type splitting of edge element vector fields we can establish asymptotic h-uniform optimality of the preconditioner defined by our auxiliary space metho
- …