FEI Solvers

After the FEI has been used to assemble the global linear system (as described in Chapter Finite Element Interface), a number of hypre solvers can be called to perform the solution. This is straightforward, if hypre’s FEI has been used. If an external FEI is employed, the user needs to link with hypre’s implementation of the LinearSystemCore class, as described in Section Using HYPRE in External FEI Implementations.

Solver parameters are specified as an array of strings, and a complete list of the available options can be found in the FEI section of the reference manual. They are passed to the FEI as in the following example:

nParams = 5;
paramStrings = new char*[nParams];
for (i = 0; i < nParams; i++) }
   paramStrings[i] = new char[100];

strcpy(paramStrings[0], "solver cg");
strcpy(paramStrings[1], "preconditioner diag");
strcpy(paramStrings[2], "maxiterations 100");
strcpy(paramStrings[3], "tolerance 1.0e-6");
strcpy(paramStrings[4], "outputLevel 1");

feiPtr -> parameters(nParams, paramStrings);

To solve the linear system of equations, we call

feiPtr -> solve(&status);

where the returned value status indicates whether the solve was successful.

Finally, the solution can be retrieved by the following function call:

feiPtr -> getBlockNodeSolution(elemBlkID, nNodes, nodeIDList,
                               solnOffsets, solnValues);

where nodeIDList is a list of nodes in element block elemBlkID, and solnOffsets[i] is the index pointing to the first location where the variables at node \(i\) is returned in solnValues.

Solvers Available Only through the FEI

While most of the solvers from the previous sections are available through the FEI interface, there are number of additional solvers and preconditioners that are accessible only through the FEI. These solvers are briefly described in this section (see also the reference manual).

Sequential and Parallel Solvers

hypre currently has many iterative solvers. There is also internally a version of the sequential SuperLU direct solver (developed at U.C. Berkeley) suitable to small problems (may be up to the size of \(10000\)). In the following we list some of these internal solvers.

  1. Additional Krylov solvers (FGMRES, TFQMR, symmetric QMR),

  2. SuperLU direct solver (sequential),

  3. SuperLU direct solver with iterative refinement (sequential),

Parallel Preconditioners

The performance of the Krylov solvers can be improved by clever selection of preconditioners. Besides those mentioned previously in this chapter, the following preconditioners are available via the LinearSystemCore interface:

  1. the modified version of MLI, which requires the finite element substructure matrices to construct the prolongation operators,

  2. parallel domain decomposition with inexact local solves (DDIlut),

  3. least-squares polynomial preconditioner,

  4. \(2 \times 2\) block preconditioner, and

  5. \(2 \times 2\) Uzawa preconditioner.

Some of these preconditioners can be tuned by a number of internal parameters modifiable by users. A description of these parameters is given in the reference manual.

Matrix Reduction

For some structural mechanics problems with multi-point constraints the discretization matrix is indefinite (eigenvalues lie in both sides of the imaginary axis). Indefinite matrices are much more difficult to solve than definite matrices. Methods have been developed to reduce these indefinite matrices to definite matrices. Two matrix reduction algorithms have been implemented in hypre, as presented in the following subsections.

Schur Complement Reduction

The incoming linear system of equations is assumed to be in the form:

\[\begin{split}\left[ \begin{array}{cc} D & B \\ B^T & 0 \end{array} \right] \left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] = \left[ \begin{array}{c} b_1 \\ b_2 \end{array} \right]\end{split}\]

where \(D\) is a diagonal matrix. After Schur complement reduction is applied, the resulting linear system becomes

\[- B^T D^{-1} B x_2 = b_2 - B^T D^{-1} b_1.\]

Slide Surface Reduction

With the presence of slide surfaces, the matrix is in the same form as in the case of Schur complement reduction. Here \(A\) represents the relationship between the master, slave, and other degrees of freedom. The matrix block \([B^T 0]\) corresponds to the constraint equations. The goal of reduction is to eliminate the constraints. As proposed by Manteuffel, the trick is to re-order the system into a \(3 \times 3\) block matrix.

\[\begin{split}\left[ \begin{array}{ccc} A_{11} & A_{12} & N \\ A_{21} & A_{22} & D \\ N_{T} & D & 0 \\ \end{array} \right] = \left[ \begin{array}{ccc} A_{11} & \hat{A}_{12} \\ \hat{A}_{21} & \hat{A}_{22}. \end{array} \right]\end{split}\]

The reduced system has the form :

\[(A_{11} - \hat{A}_{21} \hat{A}_{22}^{-1} \hat{A}_{12}) x_1 = b_1 - \hat{A}_{21} \hat{A}_{22}^{-1} b_2,\]

which is symmetric positive definite (SPD) if the original matrix is PD. In addition, \(\hat{A}_{22}^{-1}\) is easy to compute.

There are three slide surface reduction algorithms in hypre. The first follows the matrix formulation in this section. The second is similar except that it replaces the eliminated slave equations with identity rows so that the degree of freedom at each node is preserved. This is essential for certain block algorithms such as the smoothed aggregation multilevel preconditioners. The third is similar to the second except that it is more general and can be applied to problems with intersecting slide surfaces (sequential only for intersecting slide surfaces).

Other Features

To improve the efficiency of the hypre solvers, a few other features have been incorporated. We list a few of these features below :

  1. Preconditioner reuse - For multiple linear solves with matrices that are slightly perturbed from each other, oftentimes the use of the same preconditioners can save preconditioner setup times but suffer little convergence rate degradation.

  2. Projection methods - For multiple solves that use the same matrix, previous solution vectors can sometimes be used to give a better initial guess for subsequent solves. Two projection schemes have been implemented in hypre - A-conjugate projection (for SPD matrices) and minimal residual projection (for both SPD and non-SPD matrices).

  3. The sparsity pattern of the matrix is in general not destroyed after it has been loaded to an hypre matrix. But if the matrix is not to be reused, an option is provided to clean up this pattern matrix to conserve memory usage.