3 research outputs found
An error estimate for a new scheme for mean curvature motion
International audienceIn this work, we propose a new numerical scheme for the anisotropic mean curvature equation. The solution of the scheme is not unique, but for all numerical solutions, we provide an error estimate between the continuous solution and the numerical approximation. This error estimate is not optimal, but as far as we know, this is the first one for mean curvature type equation. Our scheme is also applicable to compute the solution to dislocations dynamics equation
Limits and consistency of non-local and graph approximations to the Eikonal equation
In this paper, we study a non-local approximation of the time-dependent
(local) Eikonal equation with Dirichlet-type boundary conditions, where the
kernel in the non-local problem is properly scaled. Based on the theory of
viscosity solutions, we prove existence and uniqueness of the viscosity
solutions of both the local and non-local problems, as well as regularity
properties of these solutions in time and space. We then derive error bounds
between the solution to the non-local problem and that of the local one, both
in continuous-time and Backward Euler time discretization. We then turn to
studying continuum limits of non-local problems defined on random weighted
graphs with vertices. In particular, we establish that if the kernel scale
parameter decreases at an appropriate rate as grows, then almost surely,
the solution of the problem on graphs converges uniformly to the viscosity
solution of the local problem as the time step vanishes and the number vertices
grows large