27,906 research outputs found
Convergence analysis of domain decomposition based time integrators for degenerate parabolic equations
Domain decomposition based time integrators allow the usage of parallel and
distributed hardware, making them well-suited for the temporal discretization
of parabolic systems, in general, and degenerate parabolic problems, in
particular. The latter is due to the degenerate equations' finite speed of
propagation. In this study, a rigours convergence analysis is given for such
integrators without assuming any restrictive regularity on the solutions or the
domains. The analysis is conducted by first deriving a new variational
framework for the domain decomposition, which is applicable to the two standard
degenerate examples. That is, the -Laplace and the porous medium type vector
fields. Secondly, the decomposed vector fields are restricted to the underlying
pivot space and the time integration of the parabolic problem can then be
interpreted as an operators splitting applied to a dissipative evolution
equation. The convergence results then follow by employing elements of the
approximation theory for nonlinear semigroups
Space-time balancing domain decomposition
No separate or additional fees are collected for access to or distribution of the work.In this work, we propose two-level space-time domain decomposition preconditioners for parabolic problems discretized using finite elements. They are motivated as an extension to space-time of balancing domain decomposition by constraints preconditioners. The key ingredients to be defined are the subassembled space and operator, the coarse degrees of freedom (DOFs) in which we want to enforce continuity among subdomains at the preconditioner level, and the transfer operator from the subassembled to the original finite element space. With regard to the subassembled operator, a perturbation of the time derivative is needed to end up with a well-posed preconditioner. The set of coarse DOFs includes the time average (at the space-time subdomain) of classical space constraints plus new constraints between consecutive subdomains in time. Numerical experiments show that the proposed schemes are weakly scalable in time, i.e., we can efficiently exploit increasing computational resources to solve more time steps in the same total elapsed time. Further, the scheme is also weakly space-time scalable, since it leads to asymptotically constant iterations when solving larger problems both in space and time. Excellent wall clock time weak scalability is achieved for space-time parallel solvers on some thousands of coresPeer ReviewedPostprint (published version
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
Spatial and Physical Splittings of Semilinear Parabolic Problems
Splitting methods are widely used temporal approximation schemes for parabolic partial differential equations (PDEs). These schemes may be very efficient when a problem can be naturally decomposed into multiple parts. In this thesis, splitting methods are analysed when applied to spatial splittings (partitions of the computational domain) and physical splittings (separations of physical processes) of semilinear parabolic problems. The thesis is organized into three major themes: optimal convergence order analysis, spatial splittings and a physical splitting application.In view of the first theme, temporal semi-discretizations based on splitting methods are considered. An analysis is performed which yields convergence without order under weak regularity assumptions on the solution, and convergence orders ranging up to classical for progressively more regular solutions. The analysis is performed in the framework of maximal dissipative operators, which includes a large number of parabolic problems. The temporal results are also combined with convergence studies of spatial discretizations to prove simultaneous space–time convergence orders for full discretizations.For the second theme, two spatial splitting formulations are considered. For dimension splittings each part of the formulation represents the evolution in one spatial dimension only. Thereby, multidimensional problems can be reduced to families of one-dimensional problems. For domain decomposition splittings each part represents a problem on only a smaller subdomain of the full domain of the PDE. The results of the first theme are applied to prove optimal convergence orders for splitting schemes used in conjunction with these two splitting formulations. The last theme concerns the evaluation of a physical splitting procedure in an interdisciplinary application. A model for axonal growth out of nerve cells is considered. This model features several challenges to be addressed by a successful numerical method. It consists of a linear PDE coupled to nonlinear ordinary differential equations via a moving boundary, which is part of the solution. The biological model parameters imply a wide range of scales, both in time and space. Based on a physical splitting, a tailored scheme for this model is constructed. Its robustness and efficiency are then verified by numerical experiments
- …