Characterization of the angular coefficient method on 2D and 3D piecewise smooth boundaries

Abstract

The angular coefficient method represents a valid and efficient strategy to estimate the distribution of molecules in ultra-high vacuum systems. The problem is described through a Fredholm integral equation of the second kind that is usually solved with standard numerical methods, e.g., the finite element method or the Nyström quadrature method. In this work, we aim to rigorously study the underlying integral equation in order to verify some fundamental mathematical properties and justify the application and behaviour of such numerical methods. In particular, we address to the general scenario where domains are not globally smooth. In such context, boundary corners entail poorly regular integral kernels, which in turn lead to non-Lipschitz solutions and require the adoption of non-standard analysis techniques. By introducing the concept of vacuum-connection, we can establish a methodology to prove the well-posedness of the underlying problem. Furthermore, the undermined regularity of the analytical solution and the consequent lower numerical convergence rate are proved analytically and verified through simple numerical tests.