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This paper develops and analyzes a semi-discrete and a fully discrete finiteelement method for a one-dimensional quasilinear parabolic stochastic partialdifferential equation (SPDE) which describes the stochastic mean curvature flowfor planar curves of graphs. To circumvent the difficulty caused by the lowspatial regularity of the SPDE solution, a regularization procedure is firstproposed to approximate the SPDE, and an error estimate for the regularizedproblem is derived. A semi-discrete finite element method, and a space-timefully discrete method are then proposed to approximate the solution of theregularized SPDE problem. Strong convergence with rates are established forboth, semi- and fully discrete methods. Computational experiments are providedto study the interplay of the geometric evolution and gradient type-noises.

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