A BIEM using the Trefftz test functions for solving the inverse Cauchy and source recovery problems

In this paper we develop a global domain/boundary integral equation method for the Laplace and Poisson equations, which is based on the Green׳s second identity. A derived global relation links the source term to the Dirichlet and Neumann boundary conditions into a single integral equation in terms of the Trefftz test functions. By suitably choosing the Trefftz test functions, which are not the usual Green functions as that used in the conventional boundary integral method, the present boundary integral equation method (BIEM) can find the unknown boundary conditions for the inverse Cauchy problems very well. Even under a large noise to 10% and the data over-specified in a 25% portion of the whole boundary, the recovered result is still accurate. The inverse source problems of the Poisson equation are resolved numerically by using the BIEM which is stable and effective for strongly ill-posed case with a large noise being imposed on the supplementary data.