photo of William F. Mitchell Dr. William F. Mitchell

Personal Data

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Personal Data


Applied and Computational Mathematics Division
Information Technology Laboratory
National Institute of Standards and Technology (NIST)
US Department of Commerce


Computer Scientist, Mathematical Software Group

Mailing Address

National Institute of Standards and Technology
100 Bureau Dr., Stop 8910
Gaithersburg, MD 20899-8910 USA


Email :
Phone : (301) 975-3808
FAX : (301) 975-3553


Ph.D. 1988, University of Illinois at Urbana-Champaign, Computer Science
M.S. 1983, Purdue University, Computer Science
B.S. 1977, M.S. 1978, Clarkson University, Mathematics

Curriculum Vitae

vitae (pdf 115K)



PHAML (Parallel Hierarchical Adaptive MultiLevel) is a Fortran 90 parallel (MPI message passing and/or OpenMP shared memeory) implementation of a finite element partial differential equation solver with adaptive mesh refinement and multigrid solver. PHAML is an active research project, but the current version of the software is available.

Adaptive Mesh Refinement Benchmark Problems

The development of new algorithms and computer codes for the solution of partial differential equations (PDEs) usually involves the use of proof-of-concept test problems. Such test problems have a variety of uses such as demonstrating that a new algorithm is effective, verifying that a new code is correct in the sense of achieving the theoretical order of convergence, and comparing the performance of different algorithms and codes. The purpose of this web resource is to provide a standard set of problems suitable for benchmarking and testing adaptive mesh refinement algorithms and error estimators. The problems exhibit a variety of types of singularities (e.g. point and line singularities on the boundary and in the interior), near singularities (e.g. sharp peaks, boundary layers, and wave fronts), and other difficulties.


The Zoltan Dynamic Load-Balancing Library provides critical capability to a number of parallel applications. Zoltan includes a suite of algorithms for dynamically computing partitions of problems over sets of processors; geometric, tree-based and graph-based algorithms are included. Zoltan's object-oriented interface is easy-to-use and enables Zoltan to be used by a number of different applications. Zoltan is designed to be flexible and extensible, so different algorithms can be used, compared and added easily.


f90gl is a fortran interface for Mesa, a freely distributable 3D graphics library with an API which is very similar to that of OpenGL, and GLUT, a window system independent API toolkit for writing OpenGL programs. It provides both fortran 77 and fortran 90 interfaces, and can also be used with native OpenGL implementations.


StopWatch is a Fortran 90 module for portable, easy-to-use measurement of execution time of program segments. It supports multiple watches simultaneously, each with four clocks (total cpu, user cpu, system cpu, and wall clock). StopWatch is used by inserting subroutine calls into your source code, where the subroutine calls correspond to the buttons of a stop watch.


MGGHAT (MultiGrid Galerkin Hierarchical Adaptive Triangles) is a Fortran program for the solution of second order two dimensional elliptic partial differential equations, using adaptive refinement of second, third, or fourth order elements, and multigrid solution techniques.


If you do not have the software to read the available formats, an alternate format or a paper copy of these documents will be mailed to you if requested from William Mitchell.

Performance of hp-Adaptive Strategies for 3D Elliptic Problems, submitted to Proceedings of the Computation and Information Science Conference, 2016. (preprint, pdf, 385K)

30 Years of Newest Vertex Bisection, submitted to Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2015. (preprint, pdf, 249K)

How High a Degree is High Enough for High Order Finite Elements?, Procedia Computer Science, 15, 2015, pp. 246-255. (preprint, pdf, 583K) ( link to journal) DOI 10.1016/j.procs.2015.05.235

A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations, with M. A. McClain, ACM Transactions on Mathematical Software , 41 (1), 2014. (preprint, pdf, 792K) ( link to journal) DOI 10.1145/2629459

A Collection of 2D Elliptic Problems for Testing Adaptive Grid Refinement Algorithms, Applied Mathematics and Computation, 220, 2013, pp. 350-364. (preprint, pdf, 9.8M) ( link to journal) DOI 10.1016/j.amc.2013.05.068

Resonant Control of Polar Molecules in an Optical Lattice, with T. M. Hanna, E. Tiesinga and P. S. Julienne, Physical Review A, 85 (2), 2012. arXiv:1111.0227v1, 2011. (link to journal) DOI 10.1103/PhysRevA.85.022703

A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations (long version), with M. A. McClain, NISTIR 7824, 2011. ( pdf, 33M, 215 pages)

A Collection of 2D Elliptic Problems for Testing Adaptive Algorithms, NISTIR 7668, 2010. ( pdf, 1.6M)

A Survey of hp-Adaptive Strategies for Elliptic Partial Differential Equations, with M. A. McClain, in Recent Advances in Computational and Applied Mathematics (T. E. Simos, ed.), Springer, 2011, pp. 227-258. (preprint, pdf, 16M)

The hp-Multigrid Method Applied to hp-Adaptive Refinement of Triangular Grids, Numerical Linear Algebra with Applications, 17, 2010, pp. 211-228. (preprint, pdf, 958K)

Review of "Understanding and Implementing the Finite Element Method" by Mark S. Gockenbach, SIAM Review, 49 (3), 2007, pp. 532-533. (preprint, pdf, 69K)

A Refinement-tree Based Partitioning Method for Dynamic Load Balancing with Adaptively Refined Grids, J. Par. Dist. Comput., 67 (4), 2007, pp. 417-429. (preprint, pdf, 2.5M) ( link to journal )

Effective-range Description of a Bose Gas Under Strong One- or Two-dimensional Confinement, with P. Naidon, E. Tiesinga and P. Julienne, New Journal of Physics 9 (2007) 19. ( link to journal )

PHAML User's Guide, NISTIR 7374 , 2006. (original, pdf, 3.2M ) (latest revision, pdf)

Adaptive Grid Refinement For a Model of Two Confined and Interacting Atoms, with E. Tiesinga, Applied Numerical Mathematics , 52, (2005), pp. 235-250. ( gzipped postscript, 233k)

Hamiltonian Paths Through Two- and Three-Dimensional Grids, NIST J. Res. , 110, (2005), pp. 127-136. ( gzipped postscript, 79k)

Parallel Adaptive Multilevel Methods with Full Domain Partitions, App. Num. Anal. and Comp. Math., 1, (2004), pp. 36-48. ( gzipped postscript, 286k)

The Design of a Parallel Adaptive Multi-Level Code in Fortran 90, Proceedings of the 2002 International Conference on Computational Science, 2002. ( gzipped postscript, 50k)

The Full Domain Partition Approach to Parallel Adaptive Refinement, Grid Generation and Adaptive Algorithms, IMA Volumes in Mathematics and its Applications, 113, Springer-Verlag, 1998, pp. 151-162. ( gzipped postscript, 138k)

A Parallel Multigrid Method Using the Full Domain Partition, Electronic Transactions on Numerical Analysis, 6 (1998) pp. 224-233, special issue for proceedings of the 8th Copper Mountain Conference on Multigrid Methods. ( gzipped postscript, 95k)

The Full Domain Partition Approach to Distributing Adaptive Grids, Applied Numerical Mathematics, 26 (1998) pp. 265-275, special issue for proceedings of Grid Adaptation in Computational PDEs: Theory and Applications. (gzipped postscript, 102k)

A Fortran 90 Interface for OpenGL: Revised January 1998, NISTIR 6134, 1998. ( gzipped postscript, 45k)

StopWatch User's Guide Version 1.0, NISTIR 5971, 1997. (html) (gzipped postscript, 78k)

MGGHAT User's Guide Version 1.1, NISTIR 5948, 1997. (postscript)

Optimal multilevel iterative methods for adaptive grids, SIAM J. Sci. Statist. Comput. 13 (1992), pp. 146-167.

Adaptive refinement for arbitrary finite element spaces with hierarchical bases, J. Comp. Appl. Math. 36 (1991), pp. 65-78.

A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Soft. 15 (1989), pp. 326-347.

Unified multilevel adaptive finite element methods for elliptic problems, Ph.D. thesis, Technical report UIUCDCS-R-88-1436, Department of Computer Science, University of Illinois, Urbana, IL, 1988. ( gzipped postscript, 194k) (pdf, 260K)

Collocation software for second-order elliptic partial differential equations. ACM Trans. Math. Soft. 11 (1985), pp. 379-412. (with E.N. Houstis and J.R. Rice)

More publications.


Look for me at the following conferences:

SIAM Conference on Computational Science and Engineering (SIAM CSE'17) , Atlanta, GA, February 27 - March 3, 2017.

Last change to this page: December 21, 2016
Date this page created: 1994
Contact: William Mitchell