Numerical Solution of the Poisson Equation on Domains with a Thin Layer of Random Thickness

Abstract

The present article is dedicated to the numerical solution of the Poisson equation on domains with a thin layer of different conductivity and of random thickness. By changing the boundary condition, the boundary value problem given on a random domain is transformed into a boundary value problem on a fixed domain. The randomness is then contained in the coefficients of the new boundary condition. This thin coating can be expressed by a random Robin boundary condition which yields a third order accurate solution in the scale parameter $\varepsilon$ of the layer's thickness. With the help of the Karhunen--Loeve expansion, we transform this random boundary value problem into a deterministic parametric one with a possibly high-dimensional parameter ${\bf y}$. Based on the decay of the random fluctuations of the layer's thickness, we prove rates of decay of the derivatives of the random solution with respect to this parameter ${\bf y}$ which are robust in the scale parameter $\varepsilon$. Numerical results val...