109 research outputs found
Iterative techniques for time dependent Stokes problems
AbstractIn this paper, we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent Stokes equations using a mixed finite element spatial discretization. At each time step, a two by two block system corresponding to a perturbed Stokes problem must be solved. Although there are a number of techniques for iteratively solving this type of block system, to be effective, they require a good preconditioner for the resulting pressure operator (Schur complement). In contrast to the time independent Stokes equations where the pressure operator is well conditioned, the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced (and/or the Reynolds number is increased). In this paper, we shall describe and analyze preconditioners for the resulting pressure systems. These preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and Reynolds number parameter k
A Time-Dependent Dirichlet-Neumann Method for the Heat Equation
We present a waveform relaxation version of the Dirichlet-Neumann method for
parabolic problem. Like the Dirichlet-Neumann method for steady problems, the
method is based on a non-overlapping spatial domain decomposition, and the
iteration involves subdomain solves with Dirichlet boundary conditions followed
by subdomain solves with Neumann boundary conditions. However, each subdomain
problem is now in space and time, and the interface conditions are also
time-dependent. Using a Laplace transform argument, we show for the heat
equation that when we consider finite time intervals, the Dirichlet-Neumann
method converges, similar to the case of Schwarz waveform relaxation
algorithms. The convergence rate depends on the length of the subdomains as
well as the size of the time window. In this discussion, we only stick to the
linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and
Engineering, Vol. 98, Springer-Verlag 201
Galerkin FEM for fractional order parabolic equations with initial data in
We investigate semi-discrete numerical schemes based on the standard Galerkin
and lumped mass Galerkin finite element methods for an initial-boundary value
problem for homogeneous fractional diffusion problems with non-smooth initial
data. We assume that , is a convex
polygonal (polyhedral) domain. We theoretically justify optimal order error
estimates in - and -norms for initial data in . We confirm our theoretical findings with a number of numerical tests
that include initial data being a Dirac -function supported on a
-dimensional manifold.Comment: 13 pages, 3 figure
Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids
- …