Simulations of motion by mean curvature in bounded domains, with applications to
bubble motion and grain growth, rely upon boundary conditions that are not
necessarily compatible with the equation of motion. Three closed form solutions for
the problem exist, governing translation, rotation, and expansion of a single interface,
providing the only benchmarks for algorithm verification. We derive new identities for
the translation solution. Then we estimate the accuracy of a straightforward
algorithm to recover the analytical solution for different values of the velocity
given along the boundary.
As expected, for large
the error can reach unacceptable levels especially near the boundary. We discuss
factors influencing the accuracy and propose a simple modification of the algorithm
which improves the computational accuracy.
Keywords
motion by mean curvature, grain growth, foam rheology,
algorithms, measures of accuracy
Institute of Mathematics and Physics
and Wales Institute of Mathematical and Computational
Sciences
Aberystwyth University
Aberystwyth SY23 3BZ
United Kingdom
