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

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      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. 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.

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      Most cited references 13

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      Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations

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        Flow by mean curvature of convex surfaces into spheres

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          Phase transitions and generalized motion by mean curvature

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            Author and article information

            Journal
            1303.5930

            Numerical & Computational mathematics

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