C. Vuik

GMRESR: a family of nested GMRES methods

Recently Eirola and Nevanlinna have proposed an iterativ<: solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed per it<:ration st<:p by a suitable approximation process, e.g., by GMRES itself. Our numerical experimenb indicate that this may lead to considerable savings in CPU-timc and memory requirements in typical CFD applications.

A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems

An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.

A mass‐conserving Level‐Set method for modelling of multi‐phase flows

A mass-conserving Level-Set method to model bubbly flows is presented. The method can handle high density-ratio flows with complex interface topologies, such as flows with simultaneous occurrence of bubbles and droplets. Aspects taken into account are: a sharp front (density changes abruptly), arbitrarily shaped interfaces, surface tension, buoyancy and coalescence of droplets/bubbles. Attention is paid to mass-conservation and integrity of the interface. The proposed computational method is a Level-Set method, where a Volume-of-Fluid function is used to conserve mass when the interface is advected. The aim of the method is to combine the advantages of the Level-Set and Volume-of-Fluid methods without the disadvantages. The flow is computed with a pressure correction method with the Marker-and-Cell scheme. Interface conditions are satisfied by means of the continuous surface force methodology and the jump in the density field is maintained similar to the ghost fluid method for incompressible flows

The superlinear convergence behaviour of GMRES

Abstract GMRES is a rather popular iterative method for the solution of nonsingular nonsymmetric linear systems. It is well known that GMRES often has a so-called superlinear convergence behaviour, i.e., the rate of convergence seems to improve as the iteration proceeds. For the conjugate gradients method this phenomenon has been related to a (modest) degree of convergence of the Ritz values. It has been observed in experiments that for GMRES too, changes in the convergence behaviour seem to be related to the convergence of Ritz values. In this paper we prove that as soon as eigenvalues of the original operator are sufficiently well approximated by Ritz values, GMRES from then on converges at least as fast as for a related system in which these eigenvalues (and their eigenvector components) are missing.

On the Construction of Deflation-Based Preconditioners

In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into subdomains. If grid refinement is performed, keeping the subdomain grid resolution fixed, the condition number is insensitive to the grid size. Subdomain deflation is very easy to implement and has been parallelized on a distributed memory system with only a small amount of additional communication. Numerical experiments for a steady-state convection-diffusion problem are included.

A conservative pressure-correction method for flow at all speeds

Abstract A new fully conservative Mach-uniform staggered scheme is discussed. With this scheme one can compute flow with a Mach number ranging from the incompressible limit M ↓0 up to supersonic flow M >1, with nearly uniform efficiency and accuracy. Earlier methods are based on a nonconservative discretisation of the energy equation. This results in small discrepancies in the computed shock speed for the Euler equations. The new method has similar Mach-uniform properties as the earlier methods, but is found to converge to the correct weak solution.

A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow

In this paper we compare various preconditioners for the numerical solution of partial dierential equations. We compare a coarse grid correction preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the de ated preconditioned system is always, i.e. for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the coarse grid correction. This implies that the Conjugate Gradient method applied to the de ated preconditioned system converges always faster than the Conjugate Gradient method applied to the system preconditioned by the coarse grid correction. Numerical results for porous media flows emphasize the theoretical results.

A new iterative solver for the time-harmonic wave equation

The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.

Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

A Comparison of Deflation and the Balancing Preconditioner

In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated preconditioned system is always, i.e., for all deflation vectors and all restrictions and prolongations, below the condition number of the system preconditioned by the balancing preconditioner. Even more, we establish that both preconditioners lead to almost the same spectra. The zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used. Moreover, we prove that the A-norm of the errors of the iterates built by the deflation preconditioner is always below the A-norm of the errors of the iterates built by the balancing preconditioner. Depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system. If the amount of work is equal, both preconditioners are sensitive with respect to inexact computations. Finally, we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors. Numerical results for porous media flows emphasize the theoretical results.