Communications in Mathematical Sciences

Volume 12 (2014)

Number 3

Multiscale modeling of fluctuations in stochastic elliptic PDE models of nanosensors

Pages: 401 – 421



Clemens Heitzinger (Department of Applied Mathematics and Theoretical Physics, University of Cambridge, United Kingdom; Department of Mathematics, University of Vienna, Austria; Austrian Institute of Technology, Vienna, Austria)

Christian Ringhofer (Department of Mathematics, Arizona State University, Tempe, Ariz., U.S.A.)


In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. A homogenized equation for the covariance of the solution of stochastic elliptic PDEs is derived. In addition to the homogenized equation, a rate for the covariance and variance as the cell size tends to zero is given. For the homogenized problem, an existence and uniqueness result and further properties are shown. The multiscale problem stems from the modeling of the electrostatics in nanoscale field-effect sensors, where the fluctuations arise from random charge concentrations in the cells of a boundary layer. Finally, numerical results and a numerical verification are presented.


stochastic elliptic partial differential equation, multiscale problem, homogenization, limiting problem, rate, field-effect sensor, nanowire, BioFET

2010 Mathematics Subject Classification

35B27, 35J05, 35Q92, 62P30, 82D80, 92C50

Published 26 November 2013