On an approach to deal with Neumann boundary value problems defined on uncertain domains: Numerical experiments

Neumann boundary value problems for second order elliptic equations are considered on a 2D domain whose boundary is not known and might be even non-Lipschitz. Although the domain of definition is unknown, it is assumed that (a) it contains a known domain (subdomain), (b) it is contained in a known domain (superdomain), and (c) both the subdomain and superdomain have Lipschitz boundary. To cope with the Neumann boundary condition on the unknown boundary and to properly formulate the boundary value problem (BVP), the condition has to be reformulated. A reformulated BVP is used to estimate the difference between the BVP solution on the unknown domain and the BVP solution on the known subdomain or superdomain. To evaluate the estimate, the finite element method is applied. Numerical experiments are performed to check the estimate and its response to a shrinking region of uncertainty.