Fig. 1. Notation for the one-sided problem. A domain partitioned into regions and by the interface . The normal n to the interface is defined such that it points outward from the subdomain.

Fig. 2. Two-dimensional example of a bulk partition of the domain into a finite element mesh and corresponding partition of the interface into a set of element planes.

Fig. 3. Illustration of volume fraction gradient computation. The algorithm will calculate the gradient in the center element (indicated with heavy lines). The volume fractions will be averaged for each vertex using control volumes. The dashed lines indicate the boundary of the control volume for the top-most vertex. The edges of the control volume connect element centroids-to-centroids of element edges.

Fig. 4. Illustration of input configuration to planar smoothing using stability points. The dotted line is the given straight interface, and the solid lines in each element indicate the initial interfacial segments. The filled circles are the stability points, located at the centroid of each segment.

Fig. 5. Sample PIR reconstructed interfaces. Two-dimensional curved non-spherical test on an unstructured grid (left), and triple point, planar interfaces on an unstructured grid (right).

Fig. 6. Geometry and notation for one-sided benchmark problem.

Fig. 7. Coarse mesh (left) and zoom of interface (right) for one-sided problem.

Fig. 8. Numerical approximation u^h to the solution u of the one-sided problem.

Fig. 9. Convergence in L₂ error for the one-sided problem.

Fig. 10. Geometry and notation for two-sided benchmark problem.

Fig. 11. Coarse structured mesh and zoom near interface for two-sided problem.

Fig. 12. Convergence in energy norm for the two-sided problem. Angles listed in the legend indicate the degree of random perturbation given to the interfacial normal. Error bars are indicative of the range of results obtained using various random seeds in each case.

Fig. 13. Coarse unstructured mesh (left) and corresponding convergence results (right) for two-sided problem.

Fig. 14. Evolving interface problem: (left) mesh and interface at equally-spaced time steps; (right) interface segments only.

Fig. 15. Evolving interface geometry, at equally-spaced time steps on two subsequently refined meshes.