In this paper we establish the convergence of three computational algorithms for interface motion in a multi-phase system, which incorporate bulk effects. The algorithms considered fall under the classification of thresholding schemes, in the spirit of the celebrated Merriman-BenceOsher algorithm for producing an interface moving by mean curvature. The schemes considered here all incorporate either a local force coming from an energy in the bulk, or a non-local force coming from a volume constraint. We first establish the convergence of a scheme proposed by Ruuth-Wetton for approximating volume-preserving mean-curvature flow. Next we study a scheme for the geometric flow generated by surface tension plus bulk energy. Here the limit is motion by mean curvature (MMC) plus forcing term. Last we consider a thresholding scheme for simulating grain growth in a polycrystal surrounded by air, which incorporates boundary effects on the solid-vapor interface. The limiting flow is MMC on the inner grain boundaries, and volume-preserving MMC on the solid-vapor interface.

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We provide a new proof of convergence to motion by mean curvature (MMC) for the Merriman-Bence-Osher (MBO) thresholding algorithm. The proof is elementary and does not rely on maximum principle for the scheme. The strategy is to construct a natural ansatz of the solution and then estimate the error. The proof thus also provides a convergence rate. Only some weak integrability assumptions of the heat kernel, but not its positivity, is used. Currently the result is proved in the case when smooth and classical solution of MMC exists.

In previous literature Coykendall & Maney, as well as Axtell & Stickles, have discussed the concept of irreducible divisor graphs of elements in domains and ring with zero-divisors respectively, with two different definitions. In this paper we seek to look at the irreducible divisor graphs of ring elements under a hybrid definition of the two previous ones—in hopes that this graph will reveal structure concerning irreducible divisors in rings with zero-divisors. We also compare the three graphs and examine in what respects they are related. Other graph-theoretic properties of this graph will also be discussed.

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