Direct

The node (

) is an attribute that handles settings for direct linear system solvers. Use it together with a Stationary Solver, Eigenvalue Solver, and Time-Dependent Solver, for example. The attribute can also be used together with the Coarse Solver attribute when using multigrid linear system solvers.

An alternative to the direct linear system solvers is given by iterative linear system solvers which are handled via the Iterative attribute node. Several attribute nodes for solving linear systems can be attached to an operation node, but only one can be active at any given time.

Also see Choosing the Right Linear System Solver, which describes The MUMPS Solver, The PARDISO Solver, and The SPOOLES Solver.

General

Select a linear system . Select:

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(multifrontal massively parallel sparse direct solver) (the default).

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(parallel sparse direct solver). See Ref. 3 for more information about this solver.

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(sparse object oriented linear equations solver). See Ref. 2 for more information about this solver.

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to use a dense matrix solver. The dense matrix solver stores the LU factors in a filled matrix format. It is mainly useful for boundary element (BEM) computations.

MUMPS

For it estimates how much memory the unpivoted system requires. Enter a to tell MUMPS how much more memory the pivoted system requires. The default is 1.2.

Select a : (the default automatically selected by the MUMPS solver), , , , , or .

Select the check box to control whether the solver should use a maximum weight matching strategy or not. Click to clear the check box to turn off the weight matching strategy.

The default is , which controls whether or not pivoting should be used.

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If the default is kept (), enter a number between 0 and 1. The default is 0.1. This means that in any given column, the algorithm accepts an entry as a pivot element if its absolute value is greater than or equal to the specified pivot threshold times the largest absolute value in the column.

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For , enter a value for the , which controls the minimum size of pivots (the pivot threshold). The default is 10−8.


	The perturbation strategy is not as robust as ordinary pivoting. In order to improve the solution, MUMPS uses iterative refinements.

From the list, choose to store matrix factorizations (LU factors) as blocks on disk rather than in the computer’s internal memory. Choose to not store the matrix factorizations on disk. The default setting is , which switches the storage to disk (out-of-core) if the estimated memory (for the LU factors) is exhausting the physically available memory. For the automatic option, you can specify the fraction of the physically available memory that will be used before switching to out-of-core storage in the (a value between 0 an 1). The default is 0.99; that is, the switch occurs when 99% of the physically available memory is used.

When the list is set to or , you can choose to specify how to compute the in-core memory to control the maximum amount of internal memory allowed for the blocks (stored in RAM and not on disk) using the list:

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Choose (the default) to derive the in-core memory from system data and a given formula:

(19-27)

where you can specify Mmin in the field (default 512 MB), fuse in the field (default: 0.8; that is, 80% of currently available memory), and Kint in the field (default: 3). Mtot is the total memory on the computer, and Muse is the currently used memory on the computer.

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Choose to specify the in-core memory directly in the field. The default is 512 MB.

The MUMPS out-of-core solver stores the LU factors on the hard drive. This minimizes the internal memory usage. The cost is longer solution times because it takes longer time to read and write to disk than using the internal memory.


	You can specify the temporary directory where MUMPS stores the LU factors using the -tmpdir switch. See Running COMSOL Multiphysics.

PARDISO

Select a : (the default to perform the nested dissection faster when COMSOL Multiphysics runs multithreaded), , or .

Select a to use when factorizing the matrix:

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(the default): Selects one of the two algorithms based on the type of matrix.

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: Choose this when you have many cores as it is usually faster.

Select the check box to control whether the solver should use a maximum weight matching strategy or not. Click to clear the check box to turn off the weight matching strategy.

By default the check box is not selected. Click to select and control whether to use 2-by-2 Bunch-Kaufman partial pivoting instead of 1-by-1 diagonal pivoting.

By default the check box is not selected. Click to select and run the backward and forward solves multithreaded. This mainly improves performance when there are many cores and the problem is solved several times, such as in eigenvalue computations and iterative methods.

The field controls the minimum size of pivots (the pivot threshold ε).


	To avoid pivoting, PARDISO uses a pivot perturbation strategy that tests the magnitude of the potential pivot against a constant threshold of ε = α\|PPMPSDrADcP\|∞, where P and PMPS are permutation matrices, Dr and Dc are diagonal scaling matrices, and \| · \|∞ is the infinity norm (maximum norm). If the solver encounters a tiny pivot during elimination, it sets it to sign(lii)ε\|PPMPSDrADcP\|∞. The perturbation strategy is not as robust as ordinary pivoting. In order to improve the solution, PARDISO uses iterative refinements.

Select the check box to use the Parallel Direct Sparse Solver (PARDISO) for Clusters from Intel® MKL (Math Kernel Library) when running COMSOL Multiphysics in a distributed mode.

From the list, choose to store matrix factorizations (LU factors) as blocks on disk rather than in the computer’s internal memory. Choose to not store the matrix factorizations on disk. The default setting is , which switches the storage to disk (out-of-core) if the estimated memory (for the LU factors) is exhausting the physically available memory. For the automatic option, you can specify the fraction to be stored on disk in the (a value between 0 an 1; the default is 0.99).

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Choose (the default) to derive the in-core memory from system data and a given formula:

(19-28)

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Choose to specify the in-core memory directly in the field. The default is 512 MB.

The PARDISO out-of-core solver stores the LU factors on the hard drive. This minimizes the internal memory usage. The cost is longer solution times because it takes longer time to read and write to disk than using the internal memory.


	You can specify the temporary directory where PARDISO stores the LU factors using the -tmpdir switch. See Running COMSOL Multiphysics.

SPOOLES

Select a : (the best of nested dissection and multisection), , , or .

Enter a number between 0 and 1. The default is 0.1. This means that in any given column the algorithm accepts an entry as a pivot element if its absolute value is greater than or equal to the specified pivot threshold times the largest absolute value in the column.

Error

You can control the accuracy of the solution of the linear system from the list:

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The default is , meaning that the main solver is responsible for error management. The solver checks for errors for every linear system that is solved. To avoid false termination, the main solver continues iterating until the error check passes or until the step size is smaller than about 2.2·10−14. With this setting, linear solver errors are either added to the error description if the nonlinear solver does not converge, or added as a warning if the errors persist for the converged solution.

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Choose to check for errors for every linear system that is solved. If an error occurs in the main solver, warnings originating from the error checking in the direct solver appear. The error check asserts that the relative error times a stability constant ρ is sufficiently small. This setting is useful for debugging problems with singular or near singular formulations.

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Choose for no error checking.

Use the field to manually set the stability constant ρ. The default is 400.

The check box is selected by default (except for the eigenvalue solver) so that iterative refinement is used for direct and iterative linear solvers. For linear problems (or when a nonlinear solver is not used), this means that iterative refinement is performed when the computed solution is not good enough (that is, the error check returned an error). It is possible that the refined solution is better. Iterative refinement can be a remedy for instability when solving linear systems with a solver where convergence is slow and errors might be too large, due to ill-conditioned system matrices, for example. If a nonlinear solver is used, iterative refinement is not used by default. You can often get away with intermediate linear solver steps, but if that is not the case, select the check box to use an iterative refinement. The default value in the field is 20; you can change it if needed.


	For an example using a Stationary Solver, The Blasius Boundary Layer: Application Library path COMSOL_Multiphysics/Fluid_Dynamics/blasius_boundary_layer. For an example using an Eigenvalue Solver, Isospectral Drums: Application Library path COMSOL_Multiphysics/Equation-Based/isospectral_drums.