A new convergence proof for an iterative scheme for elliptic Cauchy problems based on alternating orthogonal projections
We consider an elliptic Cauchy problem for the Poisson equation in which both Dirichlet and Neumann boundary data are prescribed only on a portion of the boundary, while no boundary conditions are available on the remaining part of the boundary. Such problems are severely ill-posed. In this work, we study the convergence of an iterative method based on the alternating imposition of Dirichlet and Neumann boundary conditions. We formulate the scheme within a Hilbert space framework, where a yet another alternation becomes apparent: each iteration can now be interpreted as an orthogonal projection onto an appropriate closed subspace. This perspective provides a transparent geometric interpretation of the algorithm and yields a concise convergence proof based on von Neumann's classical result on alternating (orthogonal) projections. In addition to elucidating the convergence of the method, this alternating projections perspective also offers a general rate of convergence in terms of the Friedrichs angle (between two closed subspaces). We point out by means of an example that such convergence rate estimate degenerates for the elliptic Cauchy problem under consideration.