We describe how some simple properties of discrete one-forms directly relate to some old and new results concerning the parameterization of 3D mesh data. Our first result is an easy proof of Tutte’s celebrated “springembedding” theorem for planar graphs, which is widely used for parameterizing meshes with the topology of a disk as a planar embedding with a convex boundary. Our second result gen...

In this dissertation, we consider new approaches to the construction of meshes, discretization, and preconditioning of the resulting algebraic systems for the diffusion equation with discontinuous coefficients. In the first part, we discuss mixed finite element approximations of the diffusion equation on general polyhedral meshes. We introduce a non-conforming approximation method for the flux ...

The non-orthogonal boundary-fitted coordinate transformation method is applied to the solution of steady three-dimensional momentum and energy equations in laminar flow to obtain temperature field and Nusselt numbers in the thermal entry region of straight ducts of different cross sectional geometries. The conservation equations originally written in Cartesian coordinates are parabolized in the...

The goal of this project is to create a quality unstructured mesh generation tool. To this end, we have developed a C++ code that combines unconstrained Delaunay triangulation with spring dynamics in order to achieve a high-quality mesh. Our code will be made publicly available as part of the RedPack software package. As a test of this code, we consider in this paper the problem of generating a...

In this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the ...

“Discrete Duality Finite Volume” schemes (DDFV for short) on general 2D meshes, in particular non conforming ones, are studied for the Stokes problem with Dirichlet boundary conditions. The DDFV method belongs to the class of staggered schemes since the components of the velocity and the pressure are approximated on different meshes. In this paper, we investigate from a numerical and theoretica...

We derive a new a posteriori error estimator for the Signorini problem. It generalizes the standard residual-type estimators for unconstrained problems in linear elasticity by additional terms at the contact boundary addressing the non-linearity. Remarkably these additional contact-related terms vanish in the case of so-called full-contact. We prove reliability and efficiency for twoand three-d...

This paper presents a variety of novel uses of space-filling curves (SFCs) for Cartesian mesh methods in CFD. While these techniques will be demonstrated using non-body-fitted Cartesian meshes, many are applicable on general body-fitted meshes – both structured and unstructured. We demonstrate the use of single O ( N log N ) SFC-based reordering to produce single-pass ( O ( N )) algorithms for ...

This research paper presents a new contiguous allocation strategy for 3D mesh-connected multicomputers. The proposed strategy maintains a list of maximal free sub-meshes and gives priority to allocating corner and boundary free sub-meshes. The goal of corner and boundary allocation is to decrease the number of leftover free sub-meshes and increase their sizes, which is expected to reduce proces...

The main goal of this paper is to discuss the numerical simulation of propagation phenomena for time harmonic electromagnetic waves by methods combining controllability and fictitious domain techniques. These methods rely on distributed Lagrangian multipliers, which allow the propagation to be simulated on an obstacle free computational region using regular finite element meshes essentially ind...