  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. Strong convergence with rates are established for
both, semi- and fully discrete methods. Computational experiments are provided
to study the interplay of the geometric evolution and gradient type-noises.
