Multiscale modeling of fluctuations in stochastic elliptic PDE models of nanosensors
Clemens Heitzinger, Christian A. Ringhofer · Communications in Mathematical Sciences · 2013
Abstract. In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. Homogenized equations for the covariance and variance of the solution of stochastic elliptic PDEs are derived. In addition to the homogenized equations, a scaling law for the covariance and variance as the cell size tends to zero is given. For the homogenized problems, existence and uniqueness results and a priori bounds are given and further properties are proven. The multiscale problem stems from the modeling of the electrostatics in nanoscale field-effect sensors, where the fluctuations arise from randomly distributed charge concentrations in the cells of a boundary layer. Finally, numerical results and a spectral approximation are presented. Key words. Stochastic elliptic partial differential equation, multiscale problem, homogenization, limiting problem, scaling law, field-effect biosensor, nanowire, BioFET. AMS subject classifications. 35B27 Homogenization; equations in media with periodic structure, 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation, 35Q92 PDEs in connection with biology and other natural sciences,