The main accomplishments and features of PHAML are:

- low and high order finite elements on triangle grids
- a novel approach to parallel data distribution (the Full Domain Partition)
*h-*,*p-*, and*hp*-adaptive mesh refinement based on newest node bisection- multiple choices for a
*posteriori*error indicators / estimators - multiple choices for
*hp*-adaptive strategies - parallel multigrid solver based on
*h*- and*p*-hierarchical basis functions - optional hooks into popular linear system solver packages (PETSc, hypre, SuperLU, MUMPS) as alternatives to the built-in multigrid solver
- a refinement-tree based partitioning method for dynamic load balancing
- optional hooks into popular partitioning packages (Zoltan, ParMETIS) as alternatives to the built-in partitioner
- solution of scalar, linear, self-adjoint, 2D, elliptic PDEs
- solution of other classes of PDEs including systems of equations (a.k.a. multiple component solutions), eigenvalue problems (using SLEPc, ARPACK, BLOPEX), and, with external looping, parabolic and nonlinear problems.
- boundary conditions: Dirichlet, natural (usually Neumann), mixed, and periodic
- arbitrary 2D connected, bounded domains, including curved boundaries and holes
- use of Fortran 90 features such as modules for data abstraction and optional arguments for simplifying calls to PHAML procedures
- message passing parallelism through MPI
- shared memory parallelism through OpenMP
- hybrid MPI/OpenMP parallelism for clusters of multicore computers
- extensive visualization capabilities using OpenGL for portability

With its wide range of features and choices, PHAML can be (and has been) used for many purposes including:

- an elliptic partial differential equation solver for scientific and engineering applications
- the development of new numerical methods and approaches to programming parallel computers
- comparative studies of different methods (linear system solvers, partitioning algorithms, adaptive strategies, etc.)
- a classroom tool for classes on numerical methods or parallel computing

**Methods**

The research performed by the PHAML project has resulted in several advances in numerical methods for the solution of PDEs on parallel computers. Further details can be found by clicking on the link for each topic.

- The full domain partition is a new approach for data distribution on parallel computers, designed to reduce the frequency with which messages are passed between processors.
- The refinement-tree based partitioning algorithm is a new approach to partitioning the grid for dynamic load balancing.
- Advances in
*h*-,*p*- and*hp*-adaptive refinement include development of the newest node bisection method for triangles, and*hp*-adaptive strategies. - Parallel adaptive grid refinement is a relatively minor modification of sequential adaptive grid refinement, thanks to the full domain partition.
- The hierarchical basis multigrid method reduces to a standard multigrid method for linear elements and uniform grids, but immediately provides an algorithm for high order elements and adaptively refined grids.
- The parallel multigrid algorithm is a modification that only requires two communication steps per cycle.

### Software

The methods developed by the PHAML project have been implemented in the research code PHAML. PHAML is written in Fortran 90 and used MPI for message passing and OpenMP for shared memory parallelism. Further details can be found by clicking on the link for each topic.

- The classes of problems that PHAML solves is much more than just Laplace's equation on a square.
- There are several interesting aspects to the implementation of PHAML.
- PHAML supports more than one parallel model.
- The graphics in PHAML are quite extensive.

### Download

PHAML version 1.19.0 can now be downloaded as the file phaml-1.19.0.tar.gz (11.3 MB) for Unix systems and MS Windows with Cygwin. When unpacked, it will place everything in a directory named phaml-1.19.0.

PHAML version 1.11.1 is also available but is not an official release. This is the code that was used for the paper *A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations*, ACM Transactions of Mathematical Software, 41 (1), 2014. It is available in case someone wants to attempt to reproduce the results in that paper.

The User's Guide is included as a pdf.file in the distribution, or it can be obtained here as a pdf file (3.9 MB). There is also a two page Quick Start guide.

Send questions, bug reports, etc. to phaml@nist.gov.

If you would like to be added to the PHAML announcement email list, send a request to phaml@nist.gov. This is a very low volume list used only for announcements concerning PHAML.

PHAML is in the public domain and not subject to copyright. Please see the LICENSE file.

**PUBLICATIONS**

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.

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

Mitchell, W.F. and McClain, M.A., **A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations**,

*ACM Transactions on Mathematical Software*, 41 (1), 2014. (preprint, pdf, 792K) ( link to journal)

Mitchell, W.F. and McClain, M.A., **A Comparison of hp-Adaptive Strategies for Elliptic Partial Differential Equations** (long version),

*NISTIR 7824*, 2011. ( pdf, 33M, 215 pages)

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

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

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

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

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

Mitchell, W.F., **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)

Mitchell, W.F.,** Adaptive Grid Refinement and Multigrid on Cluster Computers**, * Proceedings of the 15th International Parallel and Distributed Processing Symposium*, IEEE Computer Society Press, 2001. ( gzipped postscript, 200k)

Mitchell, W.F., **A Comparison of Three Fast Repartition Methods for Adaptive Grids**, * Proceedings of the Ninth SIAM Conference on Parallel Processing for Scientific Computing*, 1999. ( gzipped postscript, 50k)

Mitchell, W.F., **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, 100k)

Mitchell, W.F., **The Full Domain Partition Approach to Parallel Adaptive Refinement**, * IMA Volumes in Mathematics and its Applications*, 113, Springer-Verlag, 1998, pp. 151-162. Volume devoted to the IMA Workshop on Grid Generation and Adaptive Algorithms. ( gzipped postscript, 138k)

Mitchell, W.F.,** The Refinement-Tree Partition for Parallel Solution of Partial Differential Equations**, * NIST Journal of Research*, 103 (1998), pp. 405-414. ( gzipped postscript, 96k)

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

Mitchell, W.F., **The Full Domain Partition Approach for Parallel Multigrid on Adaptive Grids**, *Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing*, 1997. (gzipped postscript, 179k)

Mitchell, W.F., **Refinement Tree Based Partitioning for Adaptive Grids**, *Proceedings of the 7th SIAM Conference on Parallel Processing for Scientific Computing*, SIAM, 1995, pp. 587-592. (gzipped postscript, 75k)

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

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

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

Mitchell, W.F., **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)