Arbitrary topology shape reconstruction from planar cross sections
Graphical Models and Image Processing
Active Contouring of Images with Physical A-Splines
Active Contouring of Images with Physical A-Splines
A finite element approximation of linear stochastic PDEs driven by multiplicative white noise
International Journal of Computer Mathematics - Recent Advances in Computational and Applied Mathematics in Science and Engineering
Mathematics and Computers in Simulation
Stochastic Galerkin methods for elliptic interface problems with random input
Journal of Computational and Applied Mathematics
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An essential part of any boundary value problem is the domain on which the problem is defined. The domain is often given by scanning or another digital image technique with limited resolution. This leads to significant uncertainty in the domain definition. The paper focuses on the impact of the uncertainty in the domain on the Neumann boundary value problem (NBVP). It studies a scalar NBVP defined on a sequence of domains. The sequence is supposed to converge in the set sense to a limit domain. Then the respective sequence of NBVP solutions is examined. First, it is shown that the classical variational formulation is not suitable for this type of problem as even a simple NBVP on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary. A new definition of the NBVP is introduced to avoid this difficulty by means of reformulated natural boundary conditions. Then the convergence of solutions of the NBVP is demonstrated. The uniqueness of the limit solution, however, depends on the stability property of the limit domain. Finally, estimates of the difference between two NBVP solutions on two different but close domains are given.