Poincaré-Steklov’s operators and domain decomposition methods in finite dimensional spaces.

*(English)*Zbl 0683.65097
Domain decomposition methods for partial differential equations, 1st Int. Symp., Paris/France 1987, 73-112 (1988).

[For the entire collection see Zbl 0649.00019.]

The author extends his results from an earlier publication to the case of discretized elliptic partial differential equations. Namely he defines the so-called Poincaré-Steklov operators (in the literature known also as Dirichlet-Neumann mappings) for the discrete finite element elliptic problems. These operators operate on the vector space of functions defined on the interface boundaries which are used to split up the original finite element elliptic equations into a number of uncoupled systems and, after a block Gauss elimination, to reduce the original problem only on the interfaces.

These Poincaré-Steklov operators are studied in order to be used for solving the reduced problem on the interfaces by preconditioned iterative methods. The convergence properties of a number of preconditioned domain decomposition iterative methods are studied.

In general, there is a lack of proofs of some basic results. For example, the crucial observation that the constants in the estimation of the norm of the Poincaré-Steklov operators are independent of n is not proved. Such results can be found in earlier publications of P. E. Bjørstad and O. B. Widlund [SIAM J. Numer. Anal. 23, 1097-1120 (1986; Zbl 0615.65113)] and of M. Dryja [Numer. Math. 44, 153-168 (1984; Zbl 0568.65075)].

For particular finite difference meshes discrete analogues of Poincaré- Steklov operators have been used earlier by the reviewer [Lect. Notes Math. 1228, 301-314 (1986; Zbl 0636.65106)].

The author extends his results from an earlier publication to the case of discretized elliptic partial differential equations. Namely he defines the so-called Poincaré-Steklov operators (in the literature known also as Dirichlet-Neumann mappings) for the discrete finite element elliptic problems. These operators operate on the vector space of functions defined on the interface boundaries which are used to split up the original finite element elliptic equations into a number of uncoupled systems and, after a block Gauss elimination, to reduce the original problem only on the interfaces.

These Poincaré-Steklov operators are studied in order to be used for solving the reduced problem on the interfaces by preconditioned iterative methods. The convergence properties of a number of preconditioned domain decomposition iterative methods are studied.

In general, there is a lack of proofs of some basic results. For example, the crucial observation that the constants in the estimation of the norm of the Poincaré-Steklov operators are independent of n is not proved. Such results can be found in earlier publications of P. E. Bjørstad and O. B. Widlund [SIAM J. Numer. Anal. 23, 1097-1120 (1986; Zbl 0615.65113)] and of M. Dryja [Numer. Math. 44, 153-168 (1984; Zbl 0568.65075)].

For particular finite difference meshes discrete analogues of Poincaré- Steklov operators have been used earlier by the reviewer [Lect. Notes Math. 1228, 301-314 (1986; Zbl 0636.65106)].

Reviewer: P.S.Vassilevski

##### MSC:

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

65F10 | Iterative numerical methods for linear systems |

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