The Direct 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 subnodes for solving linear systems can be attached to an operation node at once. However, if a Fully Coupled solver is used 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, The SPOOLES Solver, the The Dense Matrix Solver, and The NVIDIA cuDSS® Solver.

Select a linear system Solver. Select:

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

Select a Preordering algorithm: Automatic (the default automatically selected by the MUMPS solver), Approximate minimum degree, Approximate minimum fill, Quasi-dense approximate minimum degree, Nested dissection, or Distributed nested dissection.

Select the Row preordering checkbox (selected by default) to control whether the solver should use a maximum weight matching strategy or not. Click to clear the checkbox to turn off the weight matching strategy.

Select the Reuse preordering checkbox (selected by default) to reuse the reordering of the system, which speeds up the computation but leads to a higher memory peak. If you have selected that checkbox, the Reuse sparsity pattern checkbox is available. It is selected by default to store the sparsity patterns of the assembled matrices and try to reuse them for successive assembly processes within the same solution process. In many cases, the sparsity pattern of the system matrices does not change from one nonlinear iteration or time step to the others. Reusing the sparsity pattern from the previous iteration or step can then improve the solution performance at the cost of a usually small amount of memory.

Select the Multithreaded matrix factorization checkbox (selected by default) to use multithreading during matrix factorization. If you have selected that checkbox, the Multithreaded forward and backward solve checkbox is available. It is selected by default so that the backward and forward solves run multithreaded. These two settings mainly improve performance for larger problems when there are many cores. Click to clear the checkboxes to not run the solver multithreaded.

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

Select the Block low rank factorization checkbox to make a low rank approximation of the LU factors, both when computing them and when storing them. The value for the Block low rank factorization tolerance controls the quality of the approximation. The block low rank factorization is an approximate but accurate LU-factorization method that can provide a faster factorization and potentially save memory in your models. For the compression, select a type from the Compression type list: Normal (the default) or Aggressive. The aggressive compression can potentially use less memory and be faster, at the expense of a slightly lower accuracy for the approximations of the LU factors.

From the Out-of-core list, choose On to store all matrix factorizations (LU factors) as blocks on disk rather than in the computer’s memory. The solver reads some of the blocks into memory and performs the LU-factorization on the part that is currently in memory. The blocks of data are then written back to disc and new blocks are read into memory. The size of the blocks that the solver reads from disc is controlled by the in-core memory setting. Choose Off to not store the matrix factorizations on disk. The default setting is Automatic, 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 Memory fraction for out-of-core (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 Out-of-core list is set to Automatic or On, 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 In-core memory method list:

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

Select a Scheduling method to use when factorizing the matrix:

Select the Row preordering checkbox (selected by default) to control whether the solver should use a maximum weight matching strategy or not. Click to clear the checkbox to turn off the weight matching strategy.

Select the Reuse preordering checkbox (selected by default) to reuse the reordering of the system, which speeds up the computation but leads to a higher memory peak. If you have selected that checkbox, the Reuse sparsity pattern checkbox is available. It is selected by default to store the sparsity patterns of the assembled matrices and try to reuse them for successive assembly processes within the same solution process. In many cases, the sparsity pattern of the system matrices does not change from one nonlinear iteration or time step to the others. Reusing the sparsity pattern from the previous iteration or step can then improve the solution performance at the cost of a usually small amount of memory.

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

By default, the Multithreaded forward and backward solve checkbox is selected so that the backward and forward solves run 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. Click to clear this checkbox to not run the solver multithreaded.

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

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

From the Out-of-core list, choose On to store all matrix factorizations (LU factors) as blocks on disk rather than in the computer’s memory. The solver reads some of the blocks into memory and performs the LU-factorization on the part that is currently in memory. The blocks of data are then written back to disc and new blocks are read into memory. The size of the blocks that the solver reads from disc is controlled by the in-core memory setting. Choose Off to not store the matrix factorizations on disk. The default setting is Automatic, 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 Memory fraction for out-of-core (a value between 0 an 1; the default is 0.99). If needed, the out-of-core PARDISO solver automatically increases the in-core memory that is required.

When the Out-of-core list is set to Automatic or On, 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 In-core memory method list:

Select a Preordering algorithm: Best of ND and MS (the best of nested dissection and multisection), Minimum degree, Multisection, or Nested dissection.

Enter a Pivot threshold 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.

Select the Reuse preordering checkbox (selected by default) to reuse the reordering of the system, which speeds up the computation but leads to a higher memory peak.

Select a Preordering algorithm: Automatic (the default), Nested dissection, COLAMD/Block triangular, COLAMD/Simple block triangular, or Approximate minimum degree.

If Automatic, Nested dissection, or Approximate minimum degree is selected from the Preordering algorithm list, select the Row preordering checkbox to control whether the solver should use a maximum weight matching strategy or not. Click to clear the checkbox to turn off the weight matching strategy.

If Row preordering is selected, choose an algorithm to use for weight matching from the Row preordering algorithm list: Automatic (the default automatically selected by the cuDSS solver), cuDSS Algorithm 1, cuDSS Algorithm 2, cuDSS Algorithm 3, cuDSS Algorithm 4, or cuDSS Algorithm 5 (see Ref. 4).

Choose an algorithm to use during the factorization phase from the Factorization algorithm list: Automatic (the default automatically selected by the cuDSS solver), Default, or Modified.

Enter a Pivot threshold number. The default is 1. The cuDSS 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.

From the Floating point precision list, choose Double (the default) or Single.

Enter a value for the Pivoting perturbation, which controls the minimum size of pivots (the pivot threshold ε). The default is 10−13.

From the Hybrid memory mode list, choose Automatic (the default) to let the memory limit be decided by the cuDSS solver, or choose Always or Never. Hybrid host device memory is useful when there is insufficient memory on the GPU. When you have chosen Automatic, also specify a value between 0 and 1 in the Memory fraction for hybrid mode (default: 0.8). For Automatic and Always, you can also select the checkbox beside the text field and enter a limit as an integer in the Memory limit (MB) field. The default is 4096 MB.

Select the Use hybrid compute mode checkbox to allow the solver to use both GPU and CPU for solving. Hybrid execution mode is useful when the matrix is too small to benefit from GPU computing; some of the computations are then carried out on the CPU.

Select the Use multiple GPUs checkbox to run the computation on all GPUs on the machine on which the COMSOL process is running that meet the capability requirements.

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

Use the Factor in error estimate field to manually set the constant ρ. The default is 1. See Convergence Criteria for Linear Solvers for more information.

The Iterative refinement checkbox 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 Use in nonlinear solver checkbox to use an iterative refinement. The default value in the Maximum number of refinement field is 15; you can change it if needed. By default, the Error ratio bound is set to 0.5. For both linear and nonlinear solver runs where iterative refinements are used, iterative refinements should be terminated when the error is not decreasing along the iteration. Therefore, the error ratio bound should always be smaller than 1. Because the error is computed in the L2 norm, the error ratio bound should always be greater than 0. Setting the error ratio bound to 0.5, as suggested by Ref. 7, is a more cautious approach that rarely yields significant underestimations or overestimations of errors, and it always terminates quickly compared to an error ratio bound much closer to 1. When Check error estimate is set to Automatic, a single warning, Iterative refinement triggered, appears in the Log window if the iterative refinement is triggered. When Check error estimate is set to Yes, the same warning and the number of iterative refinements applied in each linear solver call are shown in the Log window.