Explicit hard bounding functions for boundary value problems for elliptic partial differential equations

Abstract

We obtain tight super- and sub-solutions, or hard bounding functions for the probabilistic representation of the solution, to the boundary value problem for second-order elliptic partial differential equations in an explicit polynomial form. Unlike accurate approximate solutions by the existing discretization-based methods, our hard bounding polynomial functions act as pointwise 100% confidence intervals within which the solution is guaranteed to exist. Our methodology can deal with a wide range of boundary conditions even of fully mixed type with Dirichlet, Neumann and Robin components, without increasing the complexity. The quality of hard bounds can be improved by introducing piecewise polynomial functions and domain decomposition. Convergence of hard bounding functions is proved under technical conditions. Numerical results are presented throughout the paper to support our theoretical developments and to illustrate the effectiveness of our methodology.