A finite element discretization of elliptic or parabolic equations leads to large sparse linear equation systems \( Ax=b \). To solve such a system, an iterative solver may be a choice among others (see section Solvers). Kaskade 7 provides on top of the Dune solvers a CG and an Uzawa solver.
Dune::BiCGSTABSolver<LinearSpaceX>
\(A: X \rightarrow X* \) with linear space X
Pcg<LinearSpaceX,LinearSpaceX>
should preconditioners get their own subsection? Preconditioner: for operator \( A: X \rightarrow X^* \), preconditioned by \( M^{-1}: X^* \rightarrow X \) to solve \( M^{-1} A x = M^{-1} b \)
A good preconditioner for the CG algorithm depends on the problem. Very popular: incomplete factorizations ILU(0) and extensions ILU(k), ILU(t) for sparse matrices. When to use?
enum class PrecondType { NONE, JACOBI, PARTIAL, ILUT, ILUK, ARMS, INVERSE, ADDITIVESCHWARZ,
BOOMERAMG, EUCLID, TRILINOSML, SSOR, ICC0, ICC, ILUKS, HB, DIRECT };
| preconditioner | description |
|---|---|
| NONE | creates a trivial (identity) preconditioner, i.e. no preconditioning takes place |
| JACOBI | a simple single-level Jacobi preconditioner; consisting of just the diagonal \( D = diag(A) \) of the matrix \( A \), efficient for diagonally-dominant matrices |
| PARTIAL | deprecated |
| ILUT | ILU preconditioner with threshold; preconditioner from Saads ITSOL library for sparse linear systems, extends the ILU(0) factorization \( A = LU + E \) where \( L \) and \( U \) have the same nonzero structure as the lower/upper parts of \( A \), but without concept of level of fill-in, see [1] |
| ILUK | ILU preconditioner with level of fill; preconditioner from Saads ITSOL library for sparse linear systems, extends the ILU(0) factorization with fill level k, see [2] |
| ARMS | Algebraic Recursive Multilevel Solvers; preconditioner from Saads ITSOL library for sparse linear systems, based on multilevel partial elimination, see [3] |
| INVERSE | |
| ADDITIVESCHWARZ | |
| BOOMERAMG | |
| EUCLID | |
| TRILINOSML | |
| SSOR | |
| ICC0 | Incomplete Cholesky factorization by algorithm from book “Matrix Computations” by Gene Golub & Charles van Loan; create approximate cholesky factorization L by only using the sparsity pattern of A |
| ICC | |
| ILUKS | |
| HB | |
| DIRECT |
to be shifted to the reference section?
[1] Youcef Saad: “ILUT: A dual threshold incomplete LU factorization” (1994)
[2] Youcef Saad: “Iterative Methods for Sparse Linear Systems” (2003)
[3] Youcef Saad, Brian Suchomel: “ARMS: an algebraic recursive multilevel solver for general sparse linear systems” (2002)