##
**Theory and application of Steklov-Poincaré operators for boundary-value problems.**
*(English)*
Zbl 0723.65098

Applied and industrial mathematics, Proc. Symp., Venice/Italy 1989, Math. Appl., D. Reidel Publ. Co. 56, 179-203 (1991).

[For the entire collection see Zbl 0707.00016.]

This paper deals with interface operators in boundary-value problems: how to define them and how to use them to derive numerical approximations based on domain decomposition approaches.

When matching a partial differential equations set in adjoining subregions of a given physical domain, the interface operators ensure the fulfillment of transmission conditions between the different solutions. It is thus possible to reduce the overall boundary-value problem into a subproblem depending solely on the trace of the solution upon the interface. Once the solution of such a problem is available, the original solution can be reconstructed through the solution of independent boundary-value problems within each subregion.

The keystep is then how to solve effectively the interface problem, which, for linear boundary-value problems, is nothing but a linear system for the so-called capacitance matrix (the finite dimensional counterpart of the interface operator).

In this paper the interface operators (also known as Steklov-Poincaré operators) are introduced for several types of problems (elliptic equations, Stokes system, generalized Stokes system). Some of their relevant properties are presented. From the numerical point of view, as the capacitance matrix turns out to be ill-conditioned, suitable preconditioners are necessary to speed up the convergence of any iterative method one is interested to apply. Here, optimal preconditioners are obtained after the capacitance matrix is split into terms attributable to the different subregions; the effectiveness of several gradient-like iterations on the preconditioned matrix is then analyzed. Finally, an iteration-by-subdomains interpretation of the above methods is furnished.

This paper deals with interface operators in boundary-value problems: how to define them and how to use them to derive numerical approximations based on domain decomposition approaches.

When matching a partial differential equations set in adjoining subregions of a given physical domain, the interface operators ensure the fulfillment of transmission conditions between the different solutions. It is thus possible to reduce the overall boundary-value problem into a subproblem depending solely on the trace of the solution upon the interface. Once the solution of such a problem is available, the original solution can be reconstructed through the solution of independent boundary-value problems within each subregion.

The keystep is then how to solve effectively the interface problem, which, for linear boundary-value problems, is nothing but a linear system for the so-called capacitance matrix (the finite dimensional counterpart of the interface operator).

In this paper the interface operators (also known as Steklov-Poincaré operators) are introduced for several types of problems (elliptic equations, Stokes system, generalized Stokes system). Some of their relevant properties are presented. From the numerical point of view, as the capacitance matrix turns out to be ill-conditioned, suitable preconditioners are necessary to speed up the convergence of any iterative method one is interested to apply. Here, optimal preconditioners are obtained after the capacitance matrix is split into terms attributable to the different subregions; the effectiveness of several gradient-like iterations on the preconditioned matrix is then analyzed. Finally, an iteration-by-subdomains interpretation of the above methods is furnished.

Reviewer: A.Quarteroni

### MSC:

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

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

35R35 | Free boundary problems for PDEs |