Study of a finite volume scheme for the regularized mean curvature flow level set equation

Abstract

We propose a new finite volume numerical scheme for the approximation of regularised mean curvature flow level set equations, which ensures the maximum principle, and which is shown to converge to the solution of the problem. The convergence proof uses the monotonicity of the operator, in order to get the strong convergence of the approximation of the gradient. The regularisation of the original level set problem is practically meaningful and not restrictive, especially when dealing with image processing applications. Numerical examples provide indications about the accuracy of the method.

Adaptive cell-centered finite volume method for diffusion equations on a consistent quadtree grid

“…However, our consistent grid is not Cartesian, but polygonal one, possessing the properties enabling us to use the methods based on [12].…”

“…With the shapes occurring in our consistent adaptive grid too many possibilities would be tested and the algorithms hardly can be efficient. That is the reason why we have chosen the method based on [12] where we consider also representative points on finite volume edges, but not values at the corners. Then, with a help of these points we work locally, we only need access to neighbors sharing a common edge and update values in these points with a help of conservation principle.…”

“…In this work we present a new finite volume scheme working on a deformed graded quadtree grid generalizing the method for an evaluation of gradients presented in [12].…”

“…We have chosen to solve the regularized Perona-Malik equation [8,22] for which several adaptive methods have already been elaborated [3,4,9,17,18], but working on triangular or rectangular grid elements. Our method to evaluate the gradients is based on [12] dealing with similar nonlinearities in the level set curvature driven equations. We consider also representative points on finite volume edges, which are updated using the conservation principle.…”

“…Comparing with the diamond cell adaptive method [18] the advantage is that we work locally in the sense that we only need to access neighbors sharing a common edge, not a vertex only, because if the neighborhood information of the grid is not stored explicitly, finding the neighbors with a common vertex leads to too many tests and can be time consuming. In the standard approach based on [12] the represenative points are obtained as an intersection of a line segment connecting representative points of two adjacent finite volumes and their common edge which must be mutually orthogonal. Moreover this intersection must be a midpoint of the edge, but such a requirement is not fulfilled in the consistent adaptive grid.…”