Finite element approximations of the stochastic mean curvature flow of planar curves of graphs

Abstract

This paper develops and analyzes a semi-discrete and a fully discrete finite element method for a one-dimensional quasilinear parabolic stochastic partial differential equation (SPDE) which describes the stochastic mean curvature flow for planar curves of graphs. To circumvent the difficulty caused by the low spatial regularity of the SPDE solution, a regularization procedure is first proposed to approximate the SPDE, and an error estimate for the regularized problem is derived. A semi-discrete finite element method, and a space-time fully discrete method are then proposed to approximate the solution of the regularized SPDE problem. We show $$L^2$$ -convergence with rates for both, semi- and fully discrete 1methods. Computational experiments are provided to study the interplay of the geometric evolution and gradient type-noises.