The Iterative Solvers

The following section provides more detailed information about the Iterative solver types: GMRES, FGMRES, Conjugate Gradients, BiCGStab, and TFQMR.

It also discusses the Convergence Criteria for Iterative Solvers and Selecting a Preconditioner for an Iterative Linear System Solver.


	Linear in the COMSOL Multiphysics Programming Reference Manual.

Iterative Solver Types

The following information also applies to the Krylov Preconditioner attribute.


	These solvers can roughly be ordered according to their memory usage and computational time per iteration (with least memory and time first): Conjugate gradients, BiCGStab, TFQMR, GMRES, and then FGMRES. The solvers that require less memory and computational time per iteration typically are less robust and not applicable to all problem types.

GMRES Iterative Solver

This linear system solver uses the restarted GMRES (generalized minimum residual) method (see Ref. 9 and Ref. 10). This is an iterative method for general linear systems of the form Ax = b. For fast convergence it is important to use an appropriate preconditioner.

FGMRES Iterative Solver

This solver uses the restarted FGMRES (flexible generalized minimum residual) method (Ref. 12). The solver is a variant of the GMRES solver that can handle a wider class of preconditioners in a robust way. You can, for example, use any iterative solver as preconditioner for FGMRES. The downside with the method is that it uses twice as much memory as GMRES for the same number of iterations before restart. FGMRES uses right preconditioning and therefore has the same convergence criterion as right-preconditioned GMRES. If FGMRES is used together with a constant preconditioner such as the Incomplete LU preconditioner, then the FGMRES solver is identical to the right preconditioned GMRES solver.

Conjugate Gradients Iterative Solver

This solver uses the conjugate gradients iterative method (Ref. 9, Ref. 13, and Ref. 14). It is an iterative method for linear systems of the form Ax = b where the matrix A is positive definite and (Hermitian) symmetric. Sometimes the solver also works when the matrix is not positive definite, especially if it is close to positive definite. This solver uses less memory and is often faster than the GMRES solver, but it applies to a restricted set of models.

For fast convergence it is important to use an appropriate preconditioner, which should be positive definite and (Hermitian) symmetric.

BiCGStab Iterative Solver

This solver uses the biconjugate gradient stabilized iterative method (Ref. 9 and Ref. 15) for solving general linear systems of the form Ax = b. The required memory and the computational time for one iteration with BiCGStab is constant; that is, the time and memory requirements do not increase with the number of iterations as they do for GMRES. BiCGStab uses approximately the same amount of memory as GMRES uses for two iterations. Therefore, BiCGStab typically uses less memory than GMRES.

The convergence behavior of BiCGStab is often more irregular than that of GMRES. Intermediate residuals can even be orders of magnitude larger than the initial residual, which can affect the numerical accuracy as well as the rate of convergence. If the algorithm detects poor accuracy in the residual or the risk of stagnation, it restarts the iterations with the current solution as the initial guess.

In contrast to GMRES and conjugate gradients, BiCGStab uses two matrix-vector multiplications each iteration. This also requires two preconditioning steps in each iteration. Also, when using the left-preconditioned BiCGStab, an additional preconditioning step is required each iteration. That is, left-preconditioned BiCGStab requires a total of three preconditioning steps in each iteration.

TFQMR Iterative Solver

This solver uses the transpose-free quasi-minimal residual iterative method (Ref. 25 and Ref. 26) for solving general linear systems of the form Ax = b. The required memory and the computational time for one iteration with TFQMR is constant; that is, the time and memory requirements do not increase with the number of iterations as they do for GMRES. TFQMR uses approximately the same amount of memory as GMRES uses for two iterations. Therefore, TFQMR typically uses less memory than GMRES.

TFQMR is more stable than BiCGStab because it performs a minimization. It is therefore often a better alternative than BiCGStab. The convergence behavior of TFQMR is often more regular than BiCGStab but slower than GMRES. Sometimes the underlying algorithm can break down. When the algorithm detects this, it restarts the iterations with the current solution as the initial guess.

In contrast to GMRES and conjugate gradients, TFQMR uses two matrix-vector multiplications each iteration. This also requires two preconditioning steps in each iteration.

Convergence Criteria for Iterative Solvers

When you use an iterative solver, COMSOL Multiphysics estimates the error of the solution while solving. Once the error estimate is small enough, as determined by the convergence criterion

(20-16)

the software terminates the computations and returns a solution. When you use a direct solver, COMSOL Multiphysics can optionally make a check to determine if the above convergence criterion is fulfilled after the solution step. If the error criterion is not met, the solution process is stopped and an error message is given.

The definitions of M for the various solvers are:

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For MUMPS, PARDISO, and SPOOLES, M = LU, where L and U are the LU factors computed by the solver.

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When using left-preconditioning with the iterative solvers GMRES, conjugate gradients, BiCGStab, and TFQMR, M is the preconditioner matrix.

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For the remaining iterative solvers, M is the identity matrix.

The convergence criterion in Equation 20-9 states that the iterations terminate when the relative (preconditioned) residual times the factor ρ is less than a tolerance tol. For solvers where M is equal to the identity matrix, the iterations can sometimes terminate too early with an incorrect solution if the system matrix A is ill-conditioned. For solvers where M is not equal to the identity matrix, the iterations can sometimes terminate too early if M is a poor preconditioner. If the iterations terminate too early due to an ill-conditioned system matrix or a poor preconditioner, increase the factor ρ to a number of the order of the condition number for the matrix M−1A. If ρ is greater than the condition number for the matrix M−1A, the convergence criterion implies that the relative error is less than tol: |x − A−1b| < tol ·|A−1b|.

Selecting a Preconditioner for an Iterative Linear System Solver

When using an Iterative linear system solver, you must select a preconditioner. The choice of preconditioner affects the number of iterations and the solver’s eventual convergence. Preconditioning can consume more time and memory than the actual iterative solver itself. To choose a preconditioner, right-click the node and choose one of the following preconditioners from the context menu:

Table 20-7: Selecting a Preconditioner
Preconditioner	USAGE
General Framework
Multigrid — Geometric multigrid	For elliptic or parabolic systems.
Multigrid — Algebraic multigrid and smoothed aggregation AMG	For scalar problems or loosely coupled multiphysics problems of the elliptic or parabolic type.
Domain Decomposition	For large problems in a distributed-memory system or as an alternative to a direct solver.
Krylov Preconditioner	For Helmholtz problems where the mesh does not fulfill the Nyquist criteria. It can be used on the coarse multigrid level or as a smoother.
Full or Approximate Factorization or Approximate Inverse
Incomplete LU	For nonsymmetric systems (the default preconditioner).
Sparse Approximate Inverse (SAI)	For BEM methods and as a general preconditioner or smoother. It has a costly setup phase but typically shows a good parallel scalability.
Direct Preconditioner	For small fields (an ODE, for example), where a direct solver is efficient.
Pointwise General Purpose
SOR	For elliptic problems without zeros on the diagonal. Typically better than Jacobi and still rather inexpensive.
Jacobi (diagonal scaling)	For large positive definite models.
Block Gauss-Seidel General Purpose
SCGS	For fluid-flow problems with linear elements.
SOR Line	For the same problem class as for SOR but adopted to stretched/anisotropic meshes (for example, boundary layer meshes). More expensive than SOR.
Vanka	For large indefinite problems with zeros on the diagonal of the system matrix.
Vector Element Methods
Auxiliary-Space Maxwell (AMS)	For curl-curl problems stemming from stationary or time-dependent Maxwell's equations.
SOR Gauge	For ungauged vector element formulations of Magnetostatics.
SOR Vector	For large electromagnetics models using vector elements.

Each preconditioner has its own settings; to adjust them, select the preconditioner node to open its window. If you want to solve a model without a preconditioner, disable all preconditioner nodes. Normally, only one preconditioner can be active, and if you have more than one preconditioner node, an active preconditioner node becomes disabled if you enable another preconditioner. You can use multiple preconditioners in a hybrid preconditioner approach; see Hybrid Preconditioners. It is also possible to run the iterative solver without a preconditioner by not adding any preconditioner or disabling all of them.

The Incomplete LU preconditioner, which is the default preconditioner, works in a more general context than the others, but it might be impractical because of its time and memory requirements; when they work, the multigrid preconditioners are always preferable. The SOR and Jacobi diagonal-scaling preconditioners use less time and memory but only ensure convergence of the iterative solver for positive definite problems. Problems with zeros on the diagonal are efficiently preconditioned with the Vanka preconditioner. To precondition electromagnetic problems that use vector elements for a PDE containing the curl-curl operator, use the SOR Vector preconditioner.

For details about the individual preconditioners, follow the links in the table above.

Preconditioner Selection Guidelines

The physics interface selects a default preconditioner that is usually appropriate for the problem type, at least for single physics interface models. If the default does not perform well, select another one using the following guidelines:

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If the system is elliptic or parabolic (see Elliptic and Parabolic Models) use the geometric multigrid preconditioner.

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If you solve a fluid-flow problem with linear elements only, try the SCGS preconditioner. This is the default setting for most fluid flow physics interfaces.

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If you solve an indefinite problem with zeros on the diagonal of the system matrix, such as the Navier-Stokes equations, try the Vanka preconditioner or the geometric multigrid preconditioner with Vanka or Incomplete LU as the smoother. With appropriate stabilization, it is possible in many cases to use SOR or SOR Line as a GMG pre- and postsmoother instead of Vanka, which increases performance. This is the default setting in some fluid-flow interfaces.

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If the system is positive definite but so large that the other preconditioners run out of memory, try the SOR Vector as smoother.

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If you solve an electromagnetics problem using vector elements for a PDE containing the curl-curl operator, try the geometric multigrid preconditioner with the SOR vector presmoother and the SOR vector postsmoother, or try the SOR vector preconditioner. Alternatively, if the problem is real-valued stationary or time dependent, you can try the geometric multigrid (GMG) preconditioner with the SOR presmoother and the SOR postsmoother, and AMS as the coarse solver. AMS is designed for the lowest-order vector elements. For higher-order discretizations, use GMG with the option and sufficiently number of levels such that AMS could be used efficiently as a coarse solver.

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Try the Incomplete LU preconditioner, which works for all linear systems. However, it requires the tuning of the drop tolerance (or fill ratio); it can run out of memory, and in many cases it is not the most efficient preconditioner.

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If the system is elliptic or parabolic and the system is a real-valued PDE for a single solution component (that is, a scalar problem) you can alternatively try the algebraic multigrid preconditioner.

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As an alternative to the multigrid solver and the use of a direct solver, the Domain Decomposition solver can be a memory efficient alternative and is a scalable solver well suited for use in a distributed memory system.

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When you are solving a problem on a parallel computer (shared or distributed memory), you can try the SAI preconditioner or smoother due to its scalability properties.

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For small models, or as part of a hybrid preconditioner for a multiphysics model that contains a small field (such as an ODE), you can use the Direct preconditioner.

The Incomplete LU preconditioner and sometimes the multigrid preconditioners require some tuning to get fast convergence without running out of memory (see the sections about these preconditioners).

About the Relaxation Factor

The relaxation factor ω to some extent controls the stability and convergence properties of a numerical solver by shifting its eigenvalue spectrum. The optimal value for the relaxation factor can improve convergence significantly — for example, for SOR when used as a solver. However, the optimal choice is typically a subtle task with arbitrary complexity. For preconditioners and smoothers, a sophisticated choice of the relaxation factor is less important. A value ω < 1 (under-relaxation) diminishes the impact of the smoother or preconditioner by limiting the possible modification in the variable update. There is a tradeoff between stability (a small value of ω) and quick advancement in the iterative process (ω close to 1). Some smoothers for multigrid require ω < 1, but the typical default value is 1. Over-relaxation — that is, 1 < ω < 2 — might have some benefits in special situations.

Hybrid Preconditioners

For all preconditioners, you can combine the effect of multiple preconditioners, either as a preconditioner or as smoothers, coarse solvers, or domain solvers. Such hybrid preconditioners can be useful in several cases. For instance, if you need to solve a combined multiphysics problem, you can apply the appropriate smoother to each physics interface and use the combined effect in a multigrid solver, or if the physics interface requires different types of multigrid hierarchies, you can use one multigrid hierarchy for each physics interface as a preconditioner.

You activate hybrid preconditioners in the section, which is available in all preconditioner node’s windows.

Hybridization

Select the type of preconditioner from the list: Select (the default) to use the preconditioner as a single preconditioner for the solver. Select to make it possible to create a sequence of preconditioners. The hybrid preconditioner sequence is defined by all enabled preconditioner steps for the solver. For each preconditioner you can select the variables to apply the preconditioner to from the list. For models where some dependent variables represent vector fields, you can also select individual components from the list. Select (the default), or select to choose from all solution components in the list. The selection in the list overrides the selection in the list.