  We study the coupled surface and grain boundary motion in a bicrystal in the
context of the "quarter loop" geometry. Two types of physics motions are
involved in this model: motion by mean curvature and motion by surface
diffusion. The goal is finding a formulation that can describe the coupled
motion and has good numerical behavior when discretized. Two formulations are
proposed in this paper. One of them is given by a mixed order parabolic system
and the other is given by Partial Differential Algebraic Equations. The
parabolic formulation constitutes several parabolic equations which model the
two normal direction motions separately. The performance of this formulation is
good for a short time simulation. It performs even better by adding an extra
term to adjust the tangential velocity of grid points. The PDAE formulation
preserves the scaled arc length property and performs much better with no need
to add an adjusting term. Both formulations are proven to be well-posed in a
simpler setting and are solved by finite difference methods.
