Numerical Analysis of a Steepest-Descent PDE Model for Surface Relaxation below the Roughening Temperature

Abstract

Abstract. We study the numerical solution of a PDE describing the relaxation of a crystal surface to a flat facet. The PDE is a singular, nonlinear, fourth order evolution equation, which can be viewed as the gradient flow of a convex but non-smooth energy with respect to the H −1 per inner product. Our numerical scheme uses implicit discretization in time and a mixed finite-element approximation in space. The singular character of the energy is handled using regularization, combined with a primal-dual method. We study the convergence of this scheme, both theoretically and numerically.

Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model

“…This is expected from other studies in the (non-weighted) H −1 total variation flow; see e.g. [24]. Similarly to the case where g = 0, we can study the evolution numerically by using the regularized flow…”

Numerical computations of split Bregman method for fourth order total variation flow

“…We use p = 3, β = 0.25, N x = N y = 40, λ = 1.25h −4 and µ = 5h −2 . Moreover, we use the initial value u 0 (x, y) = x(x − 1)y(y − 1) − 1/36, which is considered in [26]. We can obtain the similar numerical result quite effectively by split Bregman framework.…”

Surface Relaxation Below the Roughening Temperature: Some Recent Progress and Open Questions

“…which amounts to L 2 steepest descent for the functional SE N defined by (13). Now, SE N has a unique positive critical point, because if we write v i = w 3 i then each of its two terms is a convex function of v. In addition, one can show (using energy-type inequalities) that the solution "in similarity variables" w i (s) remains strictly positive and bounded for all time.…”

“…Those in [4,26,27] use an interesting nonlinear Galerkin scheme whose numerical convergence has not yet been proved. However the convergence of a finite-element scheme for the DL equation (29) was recently examined in [13].…”