Noel J. Walkington

A Delaunay based numerical method for three dimensions: generation, formulation, and partition

We present new geometrical and numerical analysis structure theorems for the Delaunay diagram of point sets in R~ for a fixed d where the point sets arise naturally in numerical methods. In particular, we show that if the largest ratio of the circum-radius to the length of smallest edge over all simplexes in the Delaunay diagram of P, DT(P), is bounded, (called the bounded radius-edge ratio property), then DT(P) is a subgraph of a density graph, the Delaunay spheres form a k-ply system for a constant k, and that we get optimal rates of convergence for approximate solutions of Poisson’s equation constructed using control volume techniques. The density graph result implies that DT(P) has a partition of cost O(rrl ‘Id) that can be efficiently found by the geometric separator algorithm of Miller, Teng, Thurston, and Vavasis and therefore the numerical linear system defined on D7’(P) using the finite-volume method can be solved efficiently on a parallel machine (either by a director an iterative method). The constant ply structure of Delaunay spheres leads to a linear-space point location structure for these Delaunay diagrams with O(log n) time per query. Moreover, we present a new parallel algorithm for computing the Delaunay diagram for these point sets in any fixed dimension in O(log n) random parallel time and n processors. Our results show that the bounded radius-edge ratio property is desirable for well-shaped triangular meshes for numerical methods such as finite element, finite difference, and in particular, finite volume methods.

Approximation of Liquid Crystal Flows

The numerical solution of the flow of a liquid crystal governed by a particular instance of the Ericksen--Leslie equations is considered. Convergence of finite element approximations is established under appropriate regularity hypotheses, and numerical experiments exhibiting the interaction of singularities and the coupling of the director and momentum equations are presented.

Smoothing and cleaning up slivers

A sliver is a tetrahedron whose four vertices lie close to a plane and whose perpendicular projection to that plane is a convex quadrilateral with no short edge. Slivers axe both undesirable and ubiquitous in 3-dimensional Delaunay triangulations. Even when the point-set is well-spaced, slivers may result. This paper shows that such a point set permits a small perturbation whose Delaunay triangulation contains no slivers. It also gives deterministic algorithms that compute the perturbation of n points in time O(n logn) with one processor and in time O(log n) with O(n) processors.

Computation of Microstructure Utilizing Young Measure Representations

An algorithm is proposed for the solution of non-convex variational problems. In order to avoid representing highly oscillatory functions on a mesh, an associated Young measure, which characterizes such oscillations, is also approximated. Sample calculations demonstrate the viability of this approach.

Algorithms for Computing Motion by Mean Curvature

We propose a finite element algorithm for computing the motion of a surface moving by mean curvature. The algorithm uses the level set formulation so that changes in topology of the surface can be accommodated. Stability is deduced by showing that the discrete solutions satisfy both

Diffusion of fluid in a fissured medium with microstructure

L^\infty

Co-volume methods for degenerate parabolic problems

and

Error Estimates for the Discontinuous Galerkin Methods for Parabolic Equations

W^{1,1}

On the Radius-Edge Condition in the Control Volume Method

bounds. Existence of discrete solutions and connections with Brakke flows are established. Some numerical examples and application to related problems, such as the phase field equations, are also presented.

Convergence of Numerical Approximations of the Incompressible Navier-Stokes Equations with Variable Density and Viscosity

A system of quasilinear degenerate parabolic equations arising in the modeling of diffusion in a fissured medium is studied. There is one such equation in the local cell coordinates at each point of the medium, and these are coupled through a similar equation in the global coordinates. It is shown that the initial boundary value problems are well posed in the appropriate spaces.