Polynomial Meshes: Computation and Approximation

We present the software package WAM, written in Matlab, that generates Weakly Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d polynomial least squares and interpolation on compact sets with various geometries. Possible applications range from data fitting to high-order methods for PDEs

A new quasi-monte carlo technique based on nonnegative least squares and approximate Fekete points

The computation of integrals in higher dimensions and on general domains, when no explicit cubature rules are known, can be ”easily” addressed by means of the quasi- Monte Carlo method. The method, simple in its formulation, becomes computationally inefficient when the space dimension is growing and the integration domain is particularly complex. In this paper we present two new approaches to the quasi-Monte Carlo method for cubature based on nonnegative least squares and approximate Fekete points. The main idea is to use less points and especially good points for solving the system of the moments. Good points are here intended as points with good interpolation properties, due to the strict connection between interpolation and cubature. Numerical experiments show that, in average, just a tenth of the points should be used mantaining the same approximation order of the quasi-Monte Carlo method. The method has been satisfactory applied to 2 and 3-dimensional problems on quite complex domains

On the computation of sets of points with low Lebesgue constant on the unit disk

In this paper of numerical nature, we test the Lebesgue constant of several available point sets on the disk and propose new ones that enjoy low Lebesgue constant. Furthermore we extend some results in Cuyt (2012), analyzing the case of Bos arrays whose radii are nonnegative Gauss–Gegenbauer–Lobatto nodes with exponent, noticing that the optimal still allows to achieve point sets on with low Lebesgue constant for degrees. Next we introduce an algorithm that through optimization determines point sets with the best known Lebesgue constants for. Finally, we determine theoretically a point set with the best Lebesgue constant for the case

On the existence of optimal meshes in every convex domain on the plane

In this paper we study the so called optimal polynomial meshes for domains in K⊂Rd,d≥2. These meshes are discrete point sets Yn of cardinality cnd which have the property that (norm of matrix)p(norm of matrix)K≤A(norm of matrix)p(norm of matrix)Yn for every polynomial p of degree at most n with a constant A≫1 independent of n. It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d=2

A robust simplex cut-cell method for adaptive high-order discretizations of aerodynamics and multi-physics problems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Aeronautics and Astronautics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 189-199).Despite the wide use of partial differential equation (PDE) solvers, lack of automation still hinders realizing their full potential in assisting engineering analysis and design. In particular, the process of establishing a suitable mesh for a given problem often requires heavy person-in-the-loop involvement. This thesis presents work toward the development of a robust PDE solution framework that provides a reliable output prediction in a fully-automated manner. The framework consists of: a simplex cut-cell technique which allows the mesh generation process to be independent of the geometry of interest; a discontinuous Galerkin (DG) discretization which permits an easy extension to high-order accuracy; and an anisotropic output-based adaptation which improves the discretization mesh for an accurate output prediction in a fully-automated manner. Two issues are addressed that limit the automation and robustness of the existing simplex cut-cell technique in three dimensions. The first is the intersection ambiguity due to numerical precision. We introduce adaptive precision arithmetic that guarantees intersection correctness, and develop various techniques to improve the efficiency of using this arithmetic. The second is the poor quadrature quality for arbitrarily shaped elements. We propose a high-quality and efficient cut-cell quadrature rule that satisfies a quality measure we define, and demonstrate the improvement in nonlinear solver robustness using this quadrature rule. The robustness and automation of the solution framework is then demonstrated through a range of aerodynamics problems, including inviscid and laminar flows. We develop a high-order DG method with a dual-consistent output evaluation for elliptic interface problems, and extend the simplex cut-cell technique for these problems, together with a metric-optimization adaptation algorithm to handle cut elements. This solution strategy is further extended for multi-physics problems, governed by different PDEs across the interfaces. Through numerical examples, including elliptic interface problems and a conjugate heat transfer problem, high-order accuracy is demonstrated on non-interface-conforming meshes constructed by the cut-cell technique, and mesh element size and shape on each material are automatically adjusted for an accurate output prediction.by Huafei Sun.Ph. D

Polynomial approximation and cubature at approximate Fekete and Leja points of the cylinder

The paper deals with polynomial interpolation, least-square approximation and cubature of functions defined on the rectangular cylinder, K = D × [−1, 1], with D the unit disk. The nodes used for these processes are the Approximate Fekete Points (AFP) and the Discrete Leja Points (DLP) extracted from suitable Weakly Admissible Meshes (WAMs) of the cylinder. ¿From the analysis of the growth of the Lebesgue constants, approximation and cubature errors, we show that the AFP and the DLP extracted from WAM are good points for polynomial approximation and numerical integration of functions defined on the cylinder