A novel coupled complex boundary method for solving inverse source problems

In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined from additional boundary conditions. Unlike the existing methods found in the literature, which usually use some of the boundary conditions to form a boundary value problem for the elliptic partial differential equation and the remaining boundary conditions in the objective functional for optimization to determine the source term, the novel method that we propose here has coupled complex boundary conditions. We use a complex elliptic partial differential equation with a Robin boundary condition coupling the Dirichlet and Neumann boundary data, and optimize with respect to the imaginary part of the solution in the domain to determine the source term. Then, on the basis of the complex boundary value problem, Tikhonov regularization is used to obtain a stable approximate source function and the finite element method is used for discretization. Theoretical analysis is given for both the continuous model and the discrete model. Several numerical examples are provided to show the usefulness of the proposed coupled complex boundary method.