High-order Unfitted Discretizations for Partial Differential Equations Coupled with Geometric Flow

Abstract

We consider a moving free boundary problem where a diffusion equation posed on an evolving domain bounded by a smooth surface is coupled with a mean curvature flow of the bounding surface. The evolution velocity of the geometry is not a priori known but has to be determined, as a part of the problem, by the solution to the diffusion equation and the mean curvature vector of the surface. We develop and analyze new geometrically unfitted discretization methods for solving the diffusion equation and a geometric equation of the mean curvature vector at provable high orders of accuracy. We test the methods with numerical experiments which show convergence rates predicted by our a priori error estimates. With a level set function implicitly representing the geometry, we solve an advection equation of the level set domain transported by a velocity field extended from the surface. To this end, we propose two velocity extension methods and take advantage of a high-order numerical method for hyperbolic conservation laws. By unfolding the geometrically coupled bulk-surface model into three sub-models solved using the methods, we conduct proof-of-concept numerical simulations of this solution-curvature-driven moving free boundary problem.