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Motion by (weighted) mean curvature is a geometric evolution law for surfaces, representing steepest descent with respect to (an)isotropic surface energy. It has been proposed that this motion could be computed by solving the analogous evolution law using a crystalline approximation to the surface energy. We present the first convergence analysis for a numerical scheme of this type. Our treatment is restricted to one dimensional surfaces (curves in the plane) which are graphs. In this context, the scheme amounts to a new algorithm for solving quasilinear parabolic equations in one space dimension.
We prove the convergence of a hybrid discretization to the viscosity solution of the elliptic Monge-Ampere equation. The hybrid discretization uses a standard finite difference discretization in parts of the computational domain where the solution is
We propose a one-dimensional (1D) diffusion equation (heat equation) for systems in which the diffusion constant (thermal diffusivity) varies alternately with a spatial period $a$. We solve the time evolution of the field (temperature) profile from a
This paper introduces an ultra-weak space-time DPG method for the heat equation. We prove well-posedness of the variational formulation with broken test functions and verify quasi-optimality of a practical DPG scheme. Numerical experiments visualize
We study the relativistic heat equation in one space dimension. We prove a local regularity result when the initial datum is locally Lipschitz in its support. We propose a numerical scheme that captures the known features of the solutions and allows
We consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation with time-periodic condition.