Martin J. Gander

Optimized Schwarz Methods

Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. We analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants.

Optimized Schwarz Methods without Overlap for the Helmholtz Equation

The classical Schwarz method is a domain decomposition method to solve elliptic partial differential equations in parallel. Convergence is achieved through overlap of the subdomains. We study in this paper a variant of the Schwarz method which converges without overlap for the Helmholtz equation. We show that the key ingredients for such an algorithm are the transmission conditions. We derive optimal transmission conditions which lead to convergence of the algorithm in a finite number of steps. These conditions are, however, nonlocal in nature, and we introduce local approximations which we optimize for performance of the Schwarz method. This leads to an algorithm in the class of optimized Schwarz methods. We present an asymptotic analysis of the optimized Schwarz method for two types of transmission conditions, Robin conditions and transmission conditions with second order tangential derivatives. Numerical results illustrate the effectiveness of the optimized Schwarz method on a model problem and on a problem from industry.

Analysis of the Parareal Time-Parallel Time-Integration Method

The parareal algorithm is a method to solve time-dependent problems parallel in time: it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time. In this paper the relation of the parareal algorithm to space-time multigrid and multiple shooting methods is first briefly discussed. The focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals, and linear convergence for unbounded intervals.

Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation

Waveform relaxation algorithms for partial differential equations (PDEs) are traditionally obtained by discretizing the PDE in space and then splitting the discrete operator using matrix splittings. For the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach. However, the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh. Motivated by the original development of waveform relaxation in circuit simulation, where the circuits are split in the physical domain into subcircuits, we split the PDE by using overlapping domain decomposition. We prove linear convergence of the algorithm in the continuous case on an infinite time interval, at a rate depending on the size of the overlap. This result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement. The algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings. Our analysis quantifies the empirical observation by Jeltsch and Pohl [SIAM J. Sci. Comput., 16 (1995), pp. 40--49] that the convergence rate of a multisplitting algorithm depends on the overlap. Numerical results are presented which support the convergence theory.

Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods. Simply using a Krylov method is much less effective, especially when the wave number in the Helmholtz operator becomes large, and also algebraic preconditioners such as incomplete LU factorizations do not remedy the situation. Even more powerful preconditioners such as classical domain decomposition and multigrid methods fail to lead to a convergent method, and often behave differently from their usual behavior for positive definite problems. For example increasing the overlap in a classical Schwarz method degrades its performance, as does increasing the number of smoothing steps in multigrid. The purpose of this review paper is to explain why classical iterative methods fail to be effective for Helmholtz problems, and to show different avenues that have been taken to address this difficulty.

Optimal Schwarz Waveform Relaxation for the One Dimensional Wave Equation

We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for wave propagation problems with variable coefficients in one spatial dimension. We derive transmission conditions which lead to convergence of the algorithm in a number of iterations equal to the number of subdomains, independently of the length of the time interval. These optimal transmission conditions are in general nonlocal, but we show that the nonlocality depends on the time interval under consideration, and we introduce time windows to obtain optimal performance of the algorithm with local transmission conditions in the case of piecewise constant wave speed. We show that convergence in two iterations can be achieved independently of the number of subdomains in that case. The algorithm thus scales optimally with the number of subdomains, provided the time windows are chosen appropriately. For continuously varying coefficients we prove convergence of the algorithm with local transmission conditions using energy estimates. We then introduce a finite volume discretization which permits computations on nonmatching grids, and we prove convergence of the fully discrete Schwarz waveform relaxation algorithm. We finally illustrate our analysis with numerical experiments.

Optimized Schwarz Waveform Relaxation Methods for Advection Reaction Diffusion Problems

We study in this paper a new class of waveform relaxation algorithms for large systems of ordinary differential equations arising from discretizations of partial differential equations of advection reaction diffusion type. We show that the transmission conditions between the subsystems have a tremendous influence on the convergence speed of the waveform relaxation algorithms, and we identify transmission conditions with optimal performance. Since these optimal transmission conditions are expensive to use, we introduce a class of local transmission conditions of Robin type, which approximate the optimal ones and can be used at the same cost as the classical transmission conditions. We determine the transmission conditions in this class with the best performance of the associated waveform relaxation algorithm. We show that the new algorithm is well posed and converges much faster than the classical one. We illustrate our analysis with numerical experiments.

Optimized Schwarz Methods for Maxwell's Equations

Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwells equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwells equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments.

Optimization of the Hermitian and Skew-Hermitian Splitting Iteration for Saddle-Point Problems

We study the asymptotic rate of convergence of the alternating Hermitian/skew-Hermitian iteration for solving saddle-point problems arising in the discretization of elliptic partial differential equations. By a careful analysis of the iterative scheme at the continuous level we determine optimal convergence parameters for the model problem of the Poisson equation written in div-grad form. We show that the optimized convergence rate for small mesh parameter h is asymptotically 1−O(h1/2). Furthermore we show that when the splitting is used as a preconditioner for a Krylov method, a different optimization leading to two clusters in the spectrum gives an optimal, h-independent, convergence rate. The theoretical analysis is supported by numerical experiments.

A homographic best approximation problem with application to optimized Schwarz waveform relaxation

We present and study a homographic best approximation problem, which arises in the analysis of waveform relaxation algorithms with optimized transmission conditions. Its solution characterizes in each class of transmission conditions the one with the best performance of the associated waveform relaxation algorithm. We present the particular class of first order transmission conditions in detail and show that the new waveform relaxation algorithms are well posed and converge much faster than the classical one: the number of iterations to reach a certain accuracy can be orders of magnitudes smaller. We illustrate our analysis with numerical experiments.