Statistical analysis of numerical solutions to constrained phase separation problems

Abstract

<abstract><p>We compute numerical solutions to a linearly constrained phase separation problem and a nonlinearly constrained phase separation problem on compact surfaces. Results are presented for oblate and prolate ellipsoids and Cassinian ovals. We implement a finite element, phase field method to determine solutions in the form of patches that are approximately geodesic disks for some values of the parameters. Our patches are numerical solutions to diffuse interface problems, and they exhibit qualitative features of solutions to corresponding sharp interface problems that are often studied in a $ \Gamma $-convergence setting. Our use of a nonlinear conservation constraint is motivated by a desire to sharpen the interface between two distinct regions: the patch and the rest of the surface. To this end, we explore features of the patches arising in both problems. A "geodesic protocol" is implemented to generate various statistics concerning the patch that are useful for measuring patch deviation from a geodesic disk shape. We then perform the Student's $ t $-test on paired differences of these statistics to determine whether or not there is a significant statistical difference between the linear constraint and nonlinear constraint approaches. The novel use of statistical analysis to compare these two methods reveals noteworthy differences. We show that the two approaches yield significantly different results for some of the statistics. The statistical results are found to depend on both the type of geometry and the patch size in some situations. Small patches are difficult to compute numerically, but we find that the use of a nonlinear constraint aids in their computation.</p></abstract>