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The Poincaré-Steklov operator within countably normed spaces. (English) Zbl 0937.65120

Bonnet, M. (ed.) et al., Mathematical aspects of boundary element methods. Minisymposium during the IABEM 98 conference, dedicated to Vladimir Maz’ya on the occasion of his 60th birthday on 31st December 1997, Paris, France, 1998. Boca Raton, FL: Chapman & Hall/CRC. Chapman Hall/CRC Res. Notes Math. 414, 152-164 (2000).
When treating elliptic boundary value problems by domain decomposition methods the interaction between the boundary data of the subdomains plays an important role. The mapping of the Dirichlet datum on the boundary of a subdomain to its Neumann datum is called Poincaré-Steklov operator and corresponds in its discrete form to a so-called Schur complement.
For a polygonal domain \(\Omega\subset \mathbb{R}^2\) with boundary \(\Gamma\) we consider the Dirichlet problem for the Laplacian: For given trace \(u|_\Gamma\) on \(\Gamma\) find \(u\) in \(\Omega\) with \(\Delta u=0\) in \(\Omega\).
Once, the solution \(u\) to the Dirichlet problem is known, the Neumann datum \(\partial u/\partial n|_\Gamma\) on \(\Gamma\) is computable. By this mapping we formally define the Poincaré-Steklov operator \[ S: u|_\Gamma \mapsto \frac{\partial u}{\partial n}\Biggl|_\Gamma. \] In variational form this operator is given by the solution \(Su|_\Gamma\) of the problem \[ \int_\Gamma (Su|_\Gamma) v|_\Gamma ds= \int_\Omega \nabla u\nabla v dx \quad\text{for any }v\in H^1(\Omega). \] This paper deals with the mapping properties of \(S\) purely within trace spaces on \(\Gamma\). These properties ensure exponentially fast convergence of the \(h\)-\(p\) version of the Galerkin method for the inversion of \(S\).
For the entire collection see [Zbl 0924.00038].

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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