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 $p$-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

Additive domain decomposition operator splittings -- convergence analyses in a dissipative framework

We analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our analysis is conducted by first casting the domain decomposition procedure into a variational framework based on weighted Sobolev spaces. The time integration of a parabolic system can then be interpreted as an operator splitting scheme applied to an abstract evolution equation governed by a maximal dissipative vector field. By utilizing this abstract setting, we derive an optimal temporal error analysis for the two most common choices of domain decomposition based integrators. Namely, alternating direction implicit schemes and additive splitting schemes of first and second order. For the standard first-order additive splitting scheme we also extend the error analysis to semilinear evolution equations, which may only have mild solutions.Comment: Please refer to the published article for the final version which also contains numerical experiments. Version 3 and 4: Only comments added. Version 2, page 2: Clarified statement on stability issues for ADI schemes with more than two operator

A full space-time convergence order analysis of operator splittings for linear dissipative evolution equations

The Douglas--Rachford and Peaceman--Rachford splitting methods are common choices for temporal discretizations of evolution equations. In this paper we combine these methods with spatial discretizations fulfilling some easily verifiable criteria. In the setting of linear dissipative evolution equations we prove optimal convergence orders, simultaneously in time and space. We apply our abstract results to dimension splitting of a 2D diffusion problem, where a finite element method is used for spatial discretization. To conclude, the convergence results are illustrated with numerical experiments

High order splitting schemes with complex timesteps and their application in mathematical finance

High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in the Stratonovich form. in the setting of weighted spaces, the necessary analyticity of the split semigroups can easily be proved. A numerical example from interest rate theory, the CIR2 model, is considered. The numerical results are robust for drift-dominated problems and confirm our theoretical results. (C) 2013 Elsevier B.V. All rights reserved

Convergence analysis for splitting of the abstract differential Riccati equation

We consider a splitting-based approximation of the abstract differential Riccati equation in the setting of Hilbert--Schmidt operators. The Riccati equation arises in many different areas and is important within the field of optimal control. In this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit Euler scheme, under the same low regularity requirements on the initial values. For a subsequent spatial discretization, the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter. The spatial discretizations commonly lead to large-scale problems, where the use of structural properties of the solution is essential. We therefore conclude by proving that the splitting method preserves low-rank structure in the matrix-valued case. Numerical results demonstrate the validity of the convergence analysis

Dimension splitting for quasilinear parabolic equations.

In the current paper, we derive a rigorous convergence analysis for a broad range of splitting schemes applied to abstract nonlinear evolution equations, including the Lie and Peaceman-Rachford splittings. The analysis is in particular applicable to (possibly degenerate) quasilinear parabolic problems and their dimension splittings. The abstract framework is based on the theory of maximal dissipative operators, and we both give a summary of the used theory and some extensions of the classical results. The derived convergence results are illustrated by numerical experiments