Marjorie A. McClain

A Survey of hp-Adaptive Strategies for Elliptic Partial Differential Equations

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations because it can achieve a convergence rate that is exponential in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate can not simultaneously determine whether it is better to do the refinement by h or by p. Several strategies for making this determination have been proposed over the years. In this paper we summarize these strategies and demonstrate the exponential convergence rates with two classic test problems.

A Comparison of hp -Adaptive Strategies for Elliptic Partial Differential Equations

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving PDEs because it can achieve an exponential convergence rate in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h version of the finite element method, a posteriori error estimates can be used to determine where the mesh needs to be refined, but a single error estimate cannot simultaneously determine whether it is better to do the refinement by h or p. Several strategies for making this determination have been proposed over the years. These strategies are summarized, and the results of a numerical experiment to study the performance of these strategies is presented. It was found that the reference-solution-based methods are very effective, but also considerably more expensive, in terms of computation time, than other approaches. The method based on a priori knowledge is very effective when there are known point singularities. The method based on the decay rate of the expansion coefficients appears to be the best choice as a general strategy across all categories of problems, whereas many of the other strategies perform well in particular situations and are reasonable in general.

Digital Repository of Mathematical Formulae

The purpose of the NIST Digital Repository of Mathematical Formulae (DRMF) is to create a digital compendium of mathematical formulae for orthogonal polynomials and special functions (OPSF) and of associated mathematical data. The DRMF addresses needs of working mathematicians, physicists and engineers: providing a platform for publication and interaction with OPSF formulae on the web. Using MediaWiki extensions and other existing technology (such as software and macro collections developed for the NIST Digital Library of Mathematical Functions), the DRMF acts as an interactive web domain for OPSF formulae. Whereas Wikipedia and other web authoring tools manifest notions or descriptions as first class objects, the DRMF does that with mathematical formulae. See http://gw32.iu.xsede.org/index.php/Main_Page .

Semantic Preserving Bijective Mappings of Mathematical Formulae Between Document Preparation Systems and Computer Algebra Systems

Document preparation systems like Open image in new window offer the ability to render mathematical expressions as one would write these on paper. Using Open image in new window , Open image in new window , and tools generated for use in the National Institute of Standards (NIST) Digital Library of Mathematical Functions, semantically enhanced mathematical Open image in new window markup (semantic Open image in new window ) is achieved by using a semantic macro set. Computer algebra systems (CAS) such as Maple and Mathematica use alternative markup to represent mathematical expressions. By taking advantage of Youssef’s Part-of-Math tagger and CAS internal representations, we develop algorithms to translate mathematical expressions represented in semantic Open image in new window to corresponding CAS representations and vice versa. We have also developed tools for translating the entire Wolfram Encoding Continued Fraction Knowledge and University of Antwerp Continued Fractions for Special Functions datasets, for use in the NIST Digital Repository of Mathematical Formulae. The overall goal of these efforts is to provide semantically enriched standard conforming MathML representations to the public for formulae in digital mathematics libraries. These representations include presentation MathML, content MathML, generic Open image in new window , semantic Open image in new window , and now CAS representations as well.

The internal revenue service post-of-duty location modeling system - final report

This report documents a project undertaken by the National Bureau of Standards to develop a mathematical model which identifies optimal locations of Internal Revenue Service Posts-of-Duty . The mathematical model used for this problem is the uncapacitated, fixed charge, location-allocation model which minimizes travel and facility costs, given a specified level of activity. The report includes a discussion of the location problem and the mathematical model developed. Data sources identified and used are also described. Brief descriptions of the mathematical techniques used and the interactive, user-friendly computer system built to solve the problem are also provided. The system is microcomputer -based and uses menus and graphically displayed maps of tax districts for interactive inputs and solution outputs .

Performance of hp-adaptive strategies for elliptic partial differential equations

We consider the use of hp-adaptive finite element methods for the solution of elliptic partial differential equations. As with h-adaptive refinement, local error indicators can be used to determine which elements should be refined, but they are not sufficient to also determine whether an element should be refined by h or by p. A method for making that determination is called an hp-adaptive strategy. A number of strategies have been proposed, but it is not clear which ones perform best under different situations, or even if any of the strategies are good enough to be used as a general purpose solver. In this paper we present a summary of an experimental comparison of several hp-adaptive strategies. Full details can be found in [? ]. We consider the elliptic partial differential equation