Solution of the Poisson Equation with a Thin Layer of Random Thickness

The present article is dedicated to the numerical solution of the Poisson equation with a thin layer of different conductivity and of random thickness. We change the boundary condition to transform the boundary value problem given on a random domain 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 of the layer’s thickness. Based on the decay of the Karhunen-Loève expansion of the random fluctuations of the layer’s thickness, we prove rates of decay of the derivatives of the random solution with respect to the stochastic variable. They are robust in the thickness parameter and enable the use of the quasi Monte-Carlo method or of the anisotropic stochastic collocation method for the computation of the boundary value problem’s random solution. Numerical results validate our theoretical findings.