##
**Filtering factorizations. Fast solvers for large scale systems of equations.
(Filternde Zerlegungen. Schnelle Löser für große Gleichungssysteme.)**
*(German)*
Zbl 0748.65031

Teubner Skripten zur Numerik. Stuttgart: B. G. Teubner. 174 S. (1992).

The frequency filtering method (FFM) recently proposed by the author of the present monograph in his habilitation is a new, very robust and efficient technique for solving large scale finite difference and finite element schemes approximating boundary value problems for partial differential equations. The FFM belongs to the wide class of incomplete factorization techniques.

The well-known incomplete LU-factorization \(C=(L+T)T^{-1}(U+T)\) of some finite difference or finite element system matrix \(K=L+D+U\) can be defined by the condition \(Ce=Ke\), where \(e=(1,1,\dots,1)^ T\in\mathbb{R}^ n\), i.e. the operator \(C\) coincides with the operator \(K\) on the one- dimensional subspace \(S=\text{span}(e)\). Similarly, the FFM allows us to construct operators \(C\) which coincide with \(K\) on some low-dimensional subspace \(S\) spanned by certain frequency basis covering, in some sense, the whole space \(\mathbb{R}^ n\) or some subspace of interest. Thus, one can use the FFM as autonomous solver technique called smoother-corrector method by the author, as technique for the construction of preconditioners and, finally, as smoothers in multigrid methods.

The monograph consists of 5 chapters. After the introduction (Chapter 1), the author gives the algorithmic description of the methods in Chapter 2. Chapters 3 and 4 are devoted to the convergence analysis (model problem Fourier analysis in Chapter 3 and general convergence results in Chapter 4). The numerical results presented in Chapter 5 illustrate the efficiency of the FFM for a wide class of problems including symmetric, nonsymmetric and even nonlinear boundary value problems for second-order partial differential equations. These examples confirm that the FFM is comparable, or even superior to the multigrid method, at least, in the range of practical applications. Theoretically, the costs of the FFM asymptotically behaves like \(O(h^{-d}\ln(h^{-1}))\), where \(h\) and \(d\) denote the usual discretization parameter and the spatial dimension of the problem, respectively.

The monograph is very well written and can be recommended to all specialists working in the field of the numerical solution of partial differential equations.

The well-known incomplete LU-factorization \(C=(L+T)T^{-1}(U+T)\) of some finite difference or finite element system matrix \(K=L+D+U\) can be defined by the condition \(Ce=Ke\), where \(e=(1,1,\dots,1)^ T\in\mathbb{R}^ n\), i.e. the operator \(C\) coincides with the operator \(K\) on the one- dimensional subspace \(S=\text{span}(e)\). Similarly, the FFM allows us to construct operators \(C\) which coincide with \(K\) on some low-dimensional subspace \(S\) spanned by certain frequency basis covering, in some sense, the whole space \(\mathbb{R}^ n\) or some subspace of interest. Thus, one can use the FFM as autonomous solver technique called smoother-corrector method by the author, as technique for the construction of preconditioners and, finally, as smoothers in multigrid methods.

The monograph consists of 5 chapters. After the introduction (Chapter 1), the author gives the algorithmic description of the methods in Chapter 2. Chapters 3 and 4 are devoted to the convergence analysis (model problem Fourier analysis in Chapter 3 and general convergence results in Chapter 4). The numerical results presented in Chapter 5 illustrate the efficiency of the FFM for a wide class of problems including symmetric, nonsymmetric and even nonlinear boundary value problems for second-order partial differential equations. These examples confirm that the FFM is comparable, or even superior to the multigrid method, at least, in the range of practical applications. Theoretically, the costs of the FFM asymptotically behaves like \(O(h^{-d}\ln(h^{-1}))\), where \(h\) and \(d\) denote the usual discretization parameter and the spatial dimension of the problem, respectively.

The monograph is very well written and can be recommended to all specialists working in the field of the numerical solution of partial differential equations.

Reviewer: U.Langer (Chemnitz)

### MSC:

65F10 | Iterative numerical methods for linear systems |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65Y20 | Complexity and performance of numerical algorithms |