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Abstract. We consider anisotropic Allen-Cahn equations with interfacial energy induced by an anisotropic surface energy density γ. Assuming that γ is positive, positively homogeneous of degree one, strictly convex in tangential directions to the unit sphere, and sufficiently smooth, we show stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semi-implicit time discretizations of the double-well potential. In the semi-implicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer frame invariant, but depends on the orientation of the initial configuration.

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.

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We suggest hierarchical a posteriori error estimators for time-discretized Allen-Cahn and Cahn-Hilliard equations with logarithmic potential and investigate their robustness numerically. We observe that the associated effectivity ratios seem to saturate for decreasing mesh size and are almost independent of the temperature.