A high-order elliptic PDE based level set reinitialisation method using a discontinuous Galerkin discretisation

Abstract

In this paper, an efficient, high-order accurate, level set reinitialisation method is proposed, based on the elliptic reinitialisation method (Basting and Kuzmin, 2013 [1]), which is discretised spatially using the discontinuous Galerkin (DG) symmetric interior penalty method (SIPG). In order to achieve this a number of improvements have been made to the elliptic reinitialisation method including; reformulation of the underlying minimisation problem driving the solution; adoption of a Lagrange multiplier approach for enforcing a Dirichlet boundary condition on the implicit level set interface; and adoption of a narrow band approach. Numerical examples confirm the high-order accuracy of the resultant method by demonstrating experimental orders of convergence congruent with optimal convergence rates for the SIPG method, that is hp+1 and hp in the L2 and DG norms respectively. Furthermore, the degree to which the level set function satisfies the Eikonal equation improves proportionally to hp, and the often ignored homogeneous Dirichlet boundary condition on the interface is shown to be satisfied accurately with a rate of convergence of at least h2 for all polynomial orders.