Finite Element Computations of Kolmogorov–Petrovsky–Piskunov Front Speeds in Random Shear Flows in Cylinders

Abstract

We study the Kolmogorov–Petrovsky–Piskunov minimal front speeds in spatially random shear flows in cylinders of various cross sections based on the variational principle and an associated elliptic eigenvalue problem. We compare a standard finite element method and a two-scale finite element method in random front speed computations. The two-scale method iterates solutions between coarse and fine meshes and reduces the cost of the eigenvalue computation to that of a boundary value problem while maintaining the accuracy. The two-scale method saves computing time and provides accurate enough solutions. In the case of square and elliptical cross sections, our simulation shows that larger aspect ratios of domain cross sections increase the average front speeds in agreement with an asymptotic theory.