Solving Large Acoustics Problems Using Iterative Solvers

This section has some guidance for solving large acoustics problems. For smaller problems using a direct solver like MUMPS is often the best choice. For larger problems, especially in 3D, the only option is often to use an iterative method such as multigrid.


	In large models with structured mesh you can often save DOFs by changing the default Lagrange elements to serendipity elements. For more information see Lagrange and Serendipity Shape Functions.

Automatically Generated Suggestions

If the direct solver runs out of memory a simple first approach is to enable and use one of the auto generated iterative solver suggestions. A good starting point for this is to right-click the study node and select , then expand the tree under or . Predefined iterative solver suggestions are automatically generated. Per default a direct solver is used and two iterative solvers are suggested and disabled (grayed out). To turn on one of these right-click the solver and select (or press F4). The first suggestion (GMRES with GMG) uses the GMRES iterative solver with a geometric multigrid (GMG) preconditioner. This method is typically faster than the direct solver and uses less memory for large 3D models. The second suggestion (FGMRES with GMG) uses the FGMRES iterative solver with a geometric multigrid (GMG) preconditioner. This method is more robust than GMRES, especially for problems that exhibit sharp resonances. If the GMRES suggestion does not converge try the FGMRES suggestion instead. Both suggestions along with more details are described below.


	For an example that solves a pressure acoustics model using an iterative solver see: Test Bench Car Interior: Application Library path Acoustics_Module/Automotive/test_bench_car_interior


	If PMLs are present in the model solved with an iterative method, it is necessary to use the Polynomial scaling option (the default) and the recommended 8 mesh layers. This option will ensure proper convergence of the iterative methods. See the Perfectly Matched Layers (PMLs) section for further details.

Manual Suggestions and Theory

The underlying equation for many of the problems within acoustics is the Helmholtz equation. For high frequencies (or wave numbers) the matrix resulting from a finite-element discretization becomes highly indefinite. In such situations, it can be problematic to use geometric multigrid (GMG) with simple smoothers such as Jacobi or SOR (the default smoother). Fortunately, there exist robust and memory-efficient approaches that circumvent many of the difficulties associated with solving the Helmholtz equation using geometric multigrid.

When using a geometric multigrid as a linear system solver together with simple smoothers, the Nyquist criterion must be fulfilled on the coarsest mesh. If the Nyquist criterion is not satisfied, the geometric multigrid solver might not converge. One way to get around this problem is to use GMRES or FGMRES as a linear system solver with geometric multigrid as a preconditioner.

As a good starting point for modifying the solver select on the main study node and expand the tree. Go to the and add an solver node, per default it uses the GMRES method. The default preconditioner is the incomplete LU (see the subnode to the Iterative node), right-click the solver node and select . Even if the Nyquist criterion is not fulfilled for the coarse meshes of the multigrid preconditioner, such a scheme is more likely to converge. For problems with high frequencies this approach might, however, lead to a large number of iterations. Then it might be advantageous to use either:

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Geometric multigrid as a linear system solver (set the selection to ) with GMRES as a smoother. Under the node right-click the and nodes and select the with the selection to .

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FGMRES as a linear system solver (set the selection to ) with geometric multigrid as a preconditioner (where GMRES is used as a smoother, as above).

Using GMRES or FGMRES as an outer iteration and smoother removes the requirements on the coarsest mesh. When GMRES is used as a smoother for the multigrid preconditioner, FGMRES must be used for the outer iterations because such a preconditioner is not constant (see Ref. 34).

Use GMRES as a smoother only if necessary because GMRES smoothing is very time- and memory-consuming on fine meshes, especially for many smoothing steps.

When solving large acoustics problems, the following options, in increasing order of robustness and memory requirements, can be of use:

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If the Nyquist criterion is fulfilled on the coarsest mesh, try to use geometric multigrid as a linear system solver (set as preconditioner and set the linear system solver to ) with default smoothers. The default smoothers are fast and have small memory requirements.

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An option more robust than the first point is to use GMRES as a linear system solver with geometric multigrid as a preconditioner (where default SOR smoothers are used). GMRES requires memory for storing search vectors. This option can sometimes be used successfully even when the Nyquist criterion is not fulfilled on coarser meshes. Because GMRES is not used as a smoother, this option might find a solution faster than the next two options even if a large number of outer iterations are needed for convergence.

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If the above suggestion does not work, try to use geometric multigrid as a linear system solver with GMRES as a smoother.

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If the solver still has problems converging, try to use FGMRES as a linear system solver with geometric multigrid as a preconditioner (where GMRES is used as a smoother).

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Try to use as many multigrid levels as needed to produce a coarse mesh for which a direct method can solve the problem without using a substantial amount of memory.

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If the coarse mesh is still too fine for a direct solver, try using an iterative solver with 5–10 iterations as coarse solver.


	Studies and Solvers and Multigrid in the COMSOL Multiphysics Reference Manual