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

Personal Data

Personal Data


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


Retired; formerly Computer Scientist, Mathematical Software Group


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)


William F. Mitchell (Bill Mitchell) was a member of the Mathematical Software Group of the Applied and Computational Mathematics Division of the Information Technology Laboratory at the National Institute of Standards and Technology (NIST). He received his Ph.D. in computer science from the University of Illinois at Urbana-Champaign under the direction of Prof. Robert Skeel in 1988. He is known for his research in the numerical solution of partial differential equations, in particular adaptive mesh refinement (newest vertex bisection of triangles, hp-adaptive methods), multigrid methods (hierarchical basis multigrid, hp-multigrid for high order finite elements), parallel methods (full domain partition), and dynamic load balancing (refinement-tree based partition, a form of space filling curve), and his development of the Fortran 90 interface for the OpenGL graphics library (f90gl). He also collaborated with NIST scientists to apply these methods to simulations of physics applications such as the interaction of ultra-cold neutral atoms held in an optical trap, scanning electron microscope images, quantum dot structures, and crystal growth. He was an associate editor of the Journal of Numerical Analysis, Industrial and Applied Mathematics (2006-2018) and its predecessor Applied Numerical Analysis and Computational mathematics (2001-2005), was on the scientific committee of the International Conference of Numerical Analysis and Applied Mathematics (ICNAAM, 2003-2018), and organized or chaired minisymposia and technical sessions at numerous conferences. He published 50 papers and gave over 100 talks. He refereed papers for more than 30 journals and conferences, and served on review panels for the National Science Foundation (NSF) and the Department of Energy (DOE). He was a member of the Association of Computing Machinery (ACM) and the Society for Industrial and Applied Mathematics (SIAM). Dr. Mitchell retired in September 2018.



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. The final version of the PHAML 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.


The complete list is available. Selected publications are available here.

Performance of hp-Adaptive Strategies for 3D Elliptic Problems, submitted to Proceedings of the Computation and Information Science Conference, University of Thessaly Press, to appear, 2017. (preprint, pdf, 362K)

30 Years of Newest Vertex Bisection, Journal of Numerical Analysis, Industrial and Applied Mathematics, 11 (1-2), 2017, pp. 11-22. (preprint, pdf, 539K) ( link to journal)

30 Years of Newest Vertex Bisection, (short version) Proceedings of the International Conference on Numerical Analysis and Applied Mathematics , AIP Conference Proceedings 1738, 020011, 2016. (preprint, pdf, 62K) ( link to journal) DOI 10.1063/1.4951755

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)

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)

Last change to this page: September 5, 2018
Date this page created: 1994