  A new numerical method to solve an inverse source problem for the Helmholtz
equation in inhomogenous media is proposed. This method reduces the original
inverse problem to a boundary value problem for a coupled system of elliptic
PDEs, in which the unknown source function is not involved. The Dirichlet
boundary condition is given on the entire boundary of the domain of interest
and the Neumann boundary condition is given on a part of this boundary. To
solve this problem, the quasi-reversibility method is applied. Uniqueness and
existence of the minimizer are proven. A new Carleman estimate is established.
Next, the convergence of those minimizers to the exact solution is proven using
that Carleman estimate. Results of numerical tests are presented.
