Nonsmooth Schur-Newton methods for multicomponent Cahn-Hilliard systems

Abstract

We present globally convergent nonsmooth Schur-Newton methods for the solution of discrete multicomponent Cahn-Hilliard systems with logarithmic and obstacle potentials. The method solves the nonlinear set-valued saddle-point problems arising from discretization by implicit Euler methods in time and first-order finite elements in space without regularization. Efficiency and robustness of the convergence speed for vanishing temperature is illustrated by numerical experiments.

Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

“…Our solver is similar in style to the one presented in [43], and it can be extended to the case of multi-component systems as in the article. For an alternative approach to the one taken here and in [43], see, for example, [36].…”

Numerical simulation of coarsening in binary solder alloys

“…Here, it does not matter whether @u is set-or single-valued, because NSNMG relies on convexity rather than smoothness. While originally introduced for saddle point problems with obstacles [36,37], NSNMG has been meanwhile extended to more general nonlinearities with nonsmooth convex energies [33,34,40] …”

“…In light of extensive literature on modeling, analysis, numerical analysis, and fast numerical solution of multicomponent alloys and Cahn-Hilliard systems (cf., e.g., [10,22,23,40,43,46,47]) an extension of our actual computational framework to multicomponent alloys is subject of current research.…”

Numerical Approximation of Multi-Phase Penrose–Fife Systems

“…For similar problems resulting from multi-component Cahn-Hilliard systems block Gauß-Seidel-type algorithms with component-wise and vertex-wise blocking where proposed in [12] and [43], respectively. To overcome the mesh-dependence of the Gauß-Seidel approach, a nonsmooth Schur-Newton method was proposed in [29].…”

“…Spatial discretization is performed by piecewise linear finite elements with adaptive mesh refinement based on hierarchical a posteriori error estimation [30,27]. The resulting large-scale non-smooth algebraic systems are solved by non-smooth Schur-Newton multigrid (NSNMG) methods [25,29,31] exploiting again the saddle point structure of these problems. In our numerical experiments, we observe optimal order of convergence of the spatial discretization and mesh-independent, fast convergence speed of NSNMG with nested iteration.…”