Multigrid
The Multigrid solver () is used to set up a geometric multigrid (GMG) solver or an algebraic multigrid (AMG) solver. Right-click the Iterative, Krylov Preconditioner, Presmoother, Postsmoother, or Coarse Solver attribute node to add a Multigrid solver.
Linear in the COMSOL Multiphysics Programming Reference Manual
General
Select a Solver: Geometric multigrid, Algebraic multigrid, or Smoothed aggregation AMG. The smoothed aggregation AMG solver is mainly intended for linear elasticity problems when geometric multigrid cannot be used, or when classical algebraic multigrid performs poorly. The method works by clustering nodes of degrees of freedoms into aggregates based on a connection criterion. Each aggregate then becomes a new node on the next multigrid level, and the algorithm proceeds until a certain number of levels has been reached or until the number of degrees of freedoms is sufficiently small.
For either choice, enter the:
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Number of iterations. The default is 2.
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Select a Multigrid cycle: V-cycle (the default), W-cycle, or F-cycle. For Multigrid cycle, the settings are the same as for the geometric multigrid (GMG) and algebraic multigrid (AMG) solvers.
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Enter the Number of multigrid levels to generate (the default is 1 for Geometric multigrid and 5 for Algebraic multigrid).
Geometric Multigrid
For Geometric multigrid, see The Geometric Multigrid Solver/Preconditioner for more information.
The Coarse Level options are described for the Direct node.
If None is selected, no coarse mesh is used in addition to the fine mesh. This can lead to severe reduction in convergence rate but saves memory.
Algebraic Multigrid
For Algebraic multigrid, see The Algebraic Multigrid Solvers/Preconditioners for more information. In addition to the settings above, the following settings control the automatic construction of the multigrid hierarchy:
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Enter a Maximum number of DOFs at coarsest level. The default is 5000. Coarse levels are added until the number of DOFs at the coarsest level is less than the max DOFs at coarsest level or until it has reached the number of multigrid levels.
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Enter a value or use the slider to set the Quality of multigrid hierarchy. Higher quality means faster convergence at the expense of a more time consuming setup phase. For instance, if the linear solver does not converge or if it uses too many iterations, try a higher value to increase the accuracy in each iteration, meaning fewer iterations. If the algebraic multigrid algorithm runs into memory problems, try a lower value to use less memory. The range goes from 1 to 10, where 10 gives the best quality. The default is 3.
Smoothed Aggregation AMG
For Smoothed aggregation AMG, which is a version of AMG that works well alternative for CFD and other problems where GMG works well but is more robust in the sense that only one valid mesh is needed.
The following settings control the aggregation algorithm:
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Enter a Maximum number of DOFs at coarsest level. The default is 5000. Coarse levels are added until the number of DOFs at the coarsest level is less than the max DOFs at coarsest level or until it has reached the number of multigrid levels.
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The aggregation algorithm is based on a connection criterion, which you specify as a coefficient in the Strength of connections field. A node j is connected to another node i, if where ε is the strength of connection coefficient, and Aij is the submatrix of the stiffness matrix defined by the degrees of freedoms on node i and j, respectively. Loosely speaking, the strength of connection value determines how strongly the aggregation should follow the direction of anisotropy in the problem. The default value is 0.01.
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From the Null-space vectors list, choose Constant (the default) or Rigid body modes. For linear elasticity problems, always select Rigid body modes because it enhances the convergence properties significantly.
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Select the Construct prolongators componentwise check box for CFD applications and other not strongly coupled physics. It is selected by default in predefined solver suggestions for CFD.
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The Aggregates from symmetric part check box is selected by default to exclude nonsymmetric parts.
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By default, the Prolongator smoothing check box is selected for smoothing of the prolongators according to the following settings.
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Choose how to control the smoothing using the Smoothing list. The Auto option postpones the smoothing for sdim-1 levels, where sdim is the space dimension of the problem. If you choose Manual, enter the level to start smoothing at in the Start smoothing at multigrid level field.
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The final transfer operator, P, between the fine and coarse problems are smoothed by one application of Jacobi smoothing:
where ω is the Jacobi damping factor, AF is the filtered stiffness matrix, and D is the diagonal of AF. Specify ω in the Jacobi damping factor field. The default value is 2/3.
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By default, the Use filtering check box is selected. Filtering means that entries in the stiffness matrix have been dropped if they correspond to degrees of freedoms on a node that has no strong connections. Loosely speaking, filtering highlights anisotropy in the problem and results in a sparser coarse level problem..
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The Multigrid Solvers
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Multigrid Level