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Effects of uncertainties in the domain on the solution of Dirichlet boundary value problems. (English) Zbl 1016.65088

A domain with possibly non-Lipschitz boundary is defined as a limit of monotonically expanding or shrinking domains with Lipschitz boundary. A uniquely solvable Dirichlet boundary value problem (DBVP) is defined on each of the Lipschitz domains and the limit of these solutions is investigated. The limit function also solves a DBVP on the limit domain but the problem can be depend on the sequences of domains if the limit domain is unstable with respect to the DBVP.
The authors derive estimates of the difference between the respective solutions of the DBVP on two close domains, one of which is Lipschitz and the other can be unstable. Estimates for starshaped as well as rather general domains are given. Their numerical evaluation is possible and can be done in different ways.
To illustrate obtained estimates, the authors investigate an uncertain boundary value problem defined via a digital image simulation. Numerical examples support the theoretical results.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
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