##
**Numerical solution of elliptic differential equations by reduction to the interface.**
*(English)*
Zbl 1043.65128

Lecture Notes in Computational Science and Engineering 36. Berlin: Springer (ISBN 3-540-20406-7/pbk). xi, 293 p. (2004).

Let a given domain of an elliptic differential equation be divided into subdomains such that the interfaces between the subdomains belong to a finite element (FE) mesh. When the variables at the nodes in the interior of the subdomains are eliminated (by static condensation), there remains a system of equations that lives on the interfaces. This domain decomposition method is called interface element method by the authors. This name was chosen since the boundary element method (with a splitting into subdomains) leads to equations of a similar structure. – Other methods that are to be mentioned in this context are the substructuring techniques in engineering, the finite element tearing and interconnecting method, and some implementations of the mortar element method.

The first chapter provides a quick review of the theory of the finite element method. Chapter two is concerned with the Poincaré-Steklov operator that will be one main tool of the book. This operator maps the Dirichlet data of a homogeneous elliptic equation to the Neumann data of the solution. In particular an FE discretization of the operator is obtained from the classical FE equations by the elimination of the nodes in the interior. The matrix of the equations is obtained as a Schur complement. The connection with the boundary element method is described only for the continuous equations.

Chapter 3 discusses the preconditioned conjugate gradient method for the equations that live on the interfaces. Chapter 4 considers multilevel methods as space decomposition methods, i.e. as a multilevel Schwarz method. The splitting of the trace spaces is shown to be stable by referring to the splitting of the functions that live in the subdomains.

Chapter 5 focuses on anisotropic coefficients and anisotropies in the geometry. It is shown that preconditioners lead to condition numbers that grow only logarithmically with the number of variables. \(\{\)The reviewer does not see whether the constants are bounded independently of the anisotropy.\(\}\) The robustness with respect to anistropy in the geometry is elucidated by numerical examples for a scalar equation. – {If the method would work also for systems of differential equations, people in solid mechanics would no longer have trouble with locking phenomena like shear locking.}

Chapter 6 deals with the frequency filtering technique, i.e. a technique that was originally introduced by the second author for finite element equations. Chapter 7 describes a representation of the Schur complement by sums of sparse matrices for some model cases. The last two chapters apply the method to the biharmonic equation and the Stokes problem, respectively.

The concept of the book is often as follows: One starts with the variational problem on the given domain and obtains the equations for the interfaces by looking at a Schur complement. So the equations of interest and the properties are obtained in an indirect way. For this reason the reading of the book is not a simple task. The reviewer also regrets that linguistic lapses of the Russian author have not been completely eliminated by the second author.

The first chapter provides a quick review of the theory of the finite element method. Chapter two is concerned with the Poincaré-Steklov operator that will be one main tool of the book. This operator maps the Dirichlet data of a homogeneous elliptic equation to the Neumann data of the solution. In particular an FE discretization of the operator is obtained from the classical FE equations by the elimination of the nodes in the interior. The matrix of the equations is obtained as a Schur complement. The connection with the boundary element method is described only for the continuous equations.

Chapter 3 discusses the preconditioned conjugate gradient method for the equations that live on the interfaces. Chapter 4 considers multilevel methods as space decomposition methods, i.e. as a multilevel Schwarz method. The splitting of the trace spaces is shown to be stable by referring to the splitting of the functions that live in the subdomains.

Chapter 5 focuses on anisotropic coefficients and anisotropies in the geometry. It is shown that preconditioners lead to condition numbers that grow only logarithmically with the number of variables. \(\{\)The reviewer does not see whether the constants are bounded independently of the anisotropy.\(\}\) The robustness with respect to anistropy in the geometry is elucidated by numerical examples for a scalar equation. – {If the method would work also for systems of differential equations, people in solid mechanics would no longer have trouble with locking phenomena like shear locking.}

Chapter 6 deals with the frequency filtering technique, i.e. a technique that was originally introduced by the second author for finite element equations. Chapter 7 describes a representation of the Schur complement by sums of sparse matrices for some model cases. The last two chapters apply the method to the biharmonic equation and the Stokes problem, respectively.

The concept of the book is often as follows: One starts with the variational problem on the given domain and obtains the equations for the interfaces by looking at a Schur complement. So the equations of interest and the properties are obtained in an indirect way. For this reason the reading of the book is not a simple task. The reviewer also regrets that linguistic lapses of the Russian author have not been completely eliminated by the second author.

Reviewer: Dietrich Braess (Bochum)

### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

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

65F10 | Iterative numerical methods for linear systems |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

35J25 | Boundary value problems for second-order elliptic equations |

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

35J40 | Boundary value problems for higher-order elliptic equations |

35Q30 | Navier-Stokes equations |