A homogenization boundary function method for determining inaccessible boundary of a rigid inclusion for the Poisson equation

Abstract

In this paper, the problem for determining the inner boundary of the Poisson equation in an arbitrary doubly-connected plane domain is solved, which recovers an unknown inner boundary of a rigid inclusion under the over-specified Cauchy data on the accessible outer boundary. First, a homogenization function is derived to annihilate the Dirichlet and Neumann data over-specified on the outer boundary. Second, a new concept of boundary functions is introduced, which automatically satisfy the homogeneous boundary conditions on the outer boundary. Besides the lowest order elementary boundary function, other higher-order boundary functions are obtained by multiplying the elementary boundary function to the Pascal triangle. Then, by a homogenization technique we can obtain a transformed Poisson equation in a reduced doubly-connected domain in terms of the transformed variable and solve it by the domain type collocation method, whose numerical solution is expanded by a sequence of boundary functions. The nonlinear equation for determining the unknown inner boundary is derived, which is convergent fast. The accuracy and robustness of present homogenization boundary function method are assessed through five numerical examples, by comparing the exact inner boundary to the recovered one under a large noisy disturbance.