An efficient algorithm for computing anisotropic evolution by mean curvature

A number of numerical simulations of surfaces evolving by mean curvature\r\nin anisotropic materials are presented and discussed.\r\nWe approximate an evolution law based on an appropriate definition\r\nof anisotropic mean curvature.\r\nThis is first regularized by means of a relaxation procedure that\r\nproduces a reaction-diffusion equation similar to the model\r\nproposed by Allen and Cahn for the evolution of antiphase boundaries.\r\nThe successive discretization process is based on linear finite\r\nelements in space and forward differences in time.\r\nThe final algorithm also exploits the localized behaviour of the\r\nsolution using a dynamically maintained mesh that spans the sole\r\ntransition region of the relaxed solution, corresponding to a thin\r\nstrip around the moving interface.