The deal.II finite element library: Design, features, and insights

doi:10.1016/j.camwa.2020.02.022

Computers & Mathematics with Applications

Volume 81, 1 January 2021, Pages 407-422

https://doi.org/10.1016/j.camwa.2020.02.022 Get rights and content

Abstract

deal.II is a state-of-the-art finite element library focused on generality, dimension-independent programming, parallelism, and extensibility. Herein, we outline its primary design considerations and its sophisticated features such as distributed meshes, $h p$ -adaptivity, support for complex geometries, and matrix-free algorithms. But deal.II is more than just a software library: It is also a diverse and worldwide community of developers and users, as well as an educational platform. We therefore also discuss some of the technical and social challenges and lessons learned in running a large community software project over the course of two decades.

Introduction

Mathematical software has been collected in packages for almost as long as computers have been around. The first of these packages were collections of loosely connected subroutines for specific purposes. In the earliest days, most of these were related to linear algebra problems such as the solution of linear systems, or computing eigenvalues, but also to numerical integration and differentiation. Few of these packages survive to this day, but the BLAS and related LAPACK interfaces [1], [2] are still widely used, despite the fact that BLAS was developed and standardized already in the 1970s.

Since then, mathematical software has seen the emergence of ever more sophisticated and connected libraries. This includes software for sparse linear algebra in the 1980s; support for parallel sparse linear algebra based on MPI [3] in the 1990s; and, since the late 1990s and early 2000s, libraries that provide the tools to build numerical solvers for partial differential equations (PDEs) using finite element, finite difference, or finite volume methods. Many of the libraries in this category are discussed in articles in this issue.

Among the largest of the libraries supporting numerical PDE solvers is deal.II, whose architecture, feature set, user and developer community, and applications we discuss herein. The origins of this library lie in the Numerical Analysis Group at the University of Heidelberg, Germany, where a predecessor library called DEAL (short for the “Differential Equation Analysis Library”) was developed since the mid-1990s. deal.II is a re-write of DEAL using more modern software design principles; it was started in late 1997 by Wolfgang Bangerth, Ralf Hartmann, and Guido Kanschat who, at the time, were all members of the same group in Heidelberg. Since then, deal.II has grown into a truly worldwide project with more than one million lines of C＋＋ code, to which more than 250 people have contributed, and that is managed by a dedicated group of Principal Developers located at universities, research institutes, and companies across continents.

This paper discusses aspects of the deal.II project. Specifically, Section 2 is concerned with design considerations that dictate the functionality that deal.II provides. Section 3 then covers specific functionality provided within this framework. As will become apparent there, our goal is to cover essentially everything that can be provided in a generic way to codes that want to solve specific partial differential equations using the most modern aspects of the finite element method, all while supporting modern hardware. Section 4 discusses some of the lessons we have learned running a large and complex software project, while Section 5 covers how deal.II supports our views on and activities in education in the Computational Science and Engineering arena. Section 6 briefly outlines some of the complex applications that have been built atop deal.II over the years and showcases new parallel scalability results. In Section 7 we comment on the vision and future directions of the project. We conclude in Section 8.

We end this introduction by stating that an earlier review of deal.II was previously published in [4], and that individual releases and new features are discussed in a series of papers of which the most recent ones are [5], [6], [7], [8]. Specific features of deal.II, along with details of their implementation, are discussed in a large number of papers [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. For more details and an updated list, see https://dealii.org/publications.html or the summary in [8].

Section snippets

Design considerations

Any nontrivial software package needs (written or unwritten) design principles to guide its development. Such principles provide a mental backdrop for expectations on how its components are used and interact with each other. Design principles also enable users to learn a software package efficiently, build a foundation for the evolution of the software, and aid developers in gauging an appropriate and idiomatic implementation of new features.

In the following subsections, we outline the design

Features

Having discussed what we want to achieve with deal.II, let us now turn to a discussion of the features the library offers. Fig. 1 provides an overview of the biggest building blocks of deal.II and their interplay. Each box references a concept that is, in most cases, implemented in several different ways — either as classes derived from a common base class (e.g., in the case of the finite elements, mappings, and quadrature classes), as independent classes using a generic interface (as is the

Lessons learned from the development of complex software

Having discussed design goals and available functionality in the previous two sections, it may be interesting to also put all of this development into perspective: What have we learned about the development of a complex scientific software library intended for a large user community?

In [68], we have previously given some answers on what we think makes scientific software libraries successful. Let us here summarize some of the points made there and explain how they relate to deal.II

Education

As already mentioned in Sections 2.9 A tool for a large community, 2.10 A way to build a community itself, libraries such as deal.II do not exist in a vacuum. Rather, they are surrounded by user communities requiring education: Help and documentation resources at varying levels, and basic training in the underlying numerical methods and software development. But, a project such as deal.II also enables educational opportunities: It is a tool to reach communities who may be interested in

Applications

deal.II is used by hundreds or thousands of researchers in essentially every field of the sciences and engineering, as shown by the large number of publications that build on it [38] — too many to even try and summarize. Most of these publications use codes written for a specific purpose but not publicly available. However, the project website at https://www.dealii.org also links to a number of large projects built on deal.II in the geosciences, radiation transport, the material sciences, fuel

Vision and future directions

Like many volunteer projects, deal.II does not have a formal roadmap that drives development. Rather, most new features stem from individuals’ need that arise in their research. That said, we have the following aspirational goals:

•
To provide state-of-the-art tools for a broad variety of PDEs.
•
To have excellent documentation to enable anyone to use the library.
•
To enable fast and scalable algorithms from laptops to the largest supercomputers, including new architectures.
•
To create an open,

Conclusions

In this contribution, we have outlined the design criteria and functionality of the deal.II finite element library. Like many other scientific software projects, deal.II started as a small project within one lab, with no intention of reaching beyond that point; however, it has now grown into a successful and world-wide project that is used in hundreds or thousands of research projects, with nearly a dozen principal developers who spend a substantial fraction of their time on the continued

Acknowledgments

W. Bangerth was partially supported by the National Science Foundation (NSF), USA under Award OAC-1835673 as part of the Cyberinfrastructure for Sustained Scientific Innovation (CSSI), USA program; by Award DMS-1821210; by Award EAR-1925595; and by the Computational Infrastructure in Geodynamics initiative (CIG), through the National Science Foundation, USA under Awards No. EAR-0949446 and EAR-1550901 and The University of California, USA — Davis.

T. Heister was partially supported by the

References (77)

KanschatG.
Multi-level methods for discontinuous Galerkin FEM on locally refined meshes
Comput. Struct.
(2004)
KronbichlerM. et al.
A generic interface for parallel cell-based finite element operator application
Comput. & Fluids
(2012)
GiulianiN. et al.
$π$ -BEM: A flexible parallel implementation for adaptive, geometry aware, and high order boundary element methods
Adv. Eng. Softw.
(2018)
MaierM. et al.
LinearOperator – a generic, high-level expression syntax for linear algebra
Comput. Math. Appl.
(2016)
SartoriA. et al.
deal2lkit: A toolkit library for high performance programming in deal.II
SoftwareX
(2018)
GordonW.J. et al.
Transfinite mappings and their application to grid generation
Appl. Math. Comput.
(1982)
MotamarriP. et al.
DFT-FE – a massively parallel adaptive finite-element code for large-scale density functional theory calculations
Comput. Phys. Comm.
(2020)
LawsonC.L. et al.
Basic linear algebra subprograms for fortran usage
ACM Trans. Math. Software
(1979)
AndersonE. et al.
LAPACK Users’ Guide
(1999)
Message Passing Interface ForumE.
MPI: A Message-Passing Interface Standard Version 3.1
(2015)

BangerthW. et al.

deal.II – a general purpose object oriented finite element library

ACM Trans. Math. Software

(2007)

BangerthW. et al.

The deal.II library, version 8.4

J. Numer. Math.

(2016)

ArndtD. et al.

The deal.II library, version 8.5

J. Numer. Math.

(2017)

AlzettaG. et al.

The deal.II library, version 9.0

J. Numer. Math.

(2018)

ArndtD. et al.

The deal.II library, version 9.1

J. Numer. Math.

(2019)

JanssenB. et al.

Adaptive multilevel methods with local smoothing for $H^{1}$ - and $H^{curl}$ -conforming high order finite element methods

SIAM J. Sci. Comput.

(2011)

ClevengerT.C. et al.

A flexible, parallel, adaptive geometric multigrid method for FEM

(2019)

BangerthW. et al.

Algorithms and data structures for massively parallel generic adaptive finite element codes

ACM Trans. Math. Software

(2011)

BangerthW. et al.

Data structures and requirements for $h p$ finite element software

ACM Trans. Math. Software

(2009)

DavydovD. et al.

Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics

Adv. Model. Simul. Eng. Sci.

(2017)

KronbichlerM. et al.

Fast matrix-free evaluation of discontinuous Galerkin finite element operators

ACM Trans. Math. Software

(2019)

DeSimoneA. et al.

Tools for the Solution of PDEs Defined on Curved Manifolds with Deal.II

(2009)

MaierM. et al.

Linearoperator benchmarks, version 1.0.0

(2016)

TurcksinB. et al.

Workstream – a design pattern for multicore-enabled finite element computations

ACM Trans. Math. Software

(2016)

HeltaiL. et al.

Using exact geometry information in finite element computations

(2019)

LangtangenH.P. et al.

A comprehensive set of tools for solving partial differential equations; diffpack

KirkB.S. et al.

Libmesh: a C＋＋ library for parallel adaptive mesh refinement/coarsening simulations

Eng. Comput.

(2006)

W. Bangerth, Using modern features of C＋＋ for adaptive finite element methods: dimension-independent programming in...

HenningJ.L.

Session details: SPEC CPU2006 analysis

ACM SIGARCH Comput. Archit. News

(2007)

ReindersJ.

Intel Threading Building Blocks: Outfitting C＋＋ for Multi-Core Processor Parallelism

(2007)

HerouxM.A. et al.

An overview of the Trilinos project

ACM Trans. Math. Software

(2005)

HerouxM.A.

Trilinos web page

(2018)

BalayS. et al.

PETSc web page

(2018)

BalayS. et al.

PETSC Users Manual

(2018)

BursteddeC. et al.

P4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees

SIAM J. Sci. Comput.

(2011)

BursteddeC.

Parallel tree algorithms for AMR and non-standard data access

(2018)

KynchR. et al.

Resolving the sign conflict problem for hp-hexahedral Nédélec elements with application to eddy current problems

Comp. Struct.

(2016)

CuSPARSE

(2019)

Cited by (171)

A monolithic space–time temporal multirate finite element framework for interface and volume coupled problems
2024, Journal of Computational and Applied Mathematics
In this work, we propose and computationally investigate a monolithic space–time multirate scheme for coupled problems. The novelty lies in the monolithic formulation of the multirate approach as this requires a careful design of the functional framework, corresponding discretization, and implementation. Our method of choice is a tensor-product Galerkin space–time discretization. The developments are carried out for both prototype interface- and volume coupled problems such as coupled wave-heat-problems and a displacement equation coupled to Darcy flow in a poro-elastic medium. The latter is applied to the well-known Mandel’s benchmark and a three-dimensional footing problem. Detailed computational investigations and convergence analyses give evidence that our monolithic multirate framework performs well.
Quadrature of functions with endpoint singular and generalised polynomial behaviour in computational physics
2024, Computer Physics Communications
Fast and accurate numerical integration always represented a bottleneck in high-performance computational physics, especially in large and multiscale industrial simulations involving Finite (FEM) and Boundary Element Methods (BEM). The computational demand escalates significantly in problems modelled by irregular or endpoint singular behaviours which can be approximated with generalised polynomials of real degree. This is due to both the practical limitations of finite-arithmetic computations and the inefficient samples distribution of traditional Gaussian quadrature rules. We developed a non-iterative mathematical software implementing an innovative numerical quadrature which largely enhances the precision of Gauss-Legendre formulae (G-L) for integrands modelled as generalised polynomial with the optimal amount of nodes and weights capable of guaranteeing the required numerical precision. This methodology avoids to resort to more computationally expensive techniques such as adaptive or composite quadrature rules. From a theoretical point of view, the numerical method underlying this work was preliminary presented in [1] by constructing the monomial transformation itself and providing all the necessary conditions to ensure the numerical stability and exactness of the quadrature up to machine precision. The novel contribution of this work concerns the optimal implementation of said method, the extension of its applicability at run-time with different type of inputs, the provision of additional insights on its functionalities and its straightforward implementation, in particular FEM applications or other mathematical software either as an external tool or embedded suite. The open-source, cross-platform C++ library Monomial Transformation Quadrature Rule (MTQR) has been designed to be highly portable, fast and easy to integrate in larger codebases. Numerical examples in multiple physical applications showcase the improved efficiency and accuracy when compared to traditional schemes.
Program title: MTQR
CPC Library link to program files: https://doi.org/10.17632/276f78wzsw.1
Developer's repository link: https://github.com/MTQR/MTQR
Licensing provisions: GNU General Public License 3
Programming language: C++ (C++17 standard)
Supplementary material: User manual (for installation and execution)
Nature of problem: Accuracy and time of execution of the current implementations of high-precision numerical integration routines for singular and irregular integrands modelled by generalised polynomials are restricted by: limitations of the floating-point (f.p.) finite-arithmetic of the machine; inability of classical Gaussian quadrature rules to efficiently capture irregular behaviours or end-point singularities using an optimal number of samples; relying on significantly expensive techniques as adaptive or composite quadrature rules that severely increase the number of steps necessary to converge to the desired accuracy threshold. However by precisely manipulating the G-L samples using an ad-hoc monomial transformation we achieve a one-shot, non-iterative, machine-precise quadrature rule with straightforward scalability in higher dimensions. The advantages brought by a non-adaptive technique are greatly emphasised whenever the problem on hand is characterised by a numerous set of integrand functions that behave similarly to sets of generalised polynomials.
Solution method: The underlying algorithm is an application of a monomial transformation of (real) order to the $n_{min} \in N$ quadrature samples (nodes and weights) of the G-L rule. The order of the transformation depends on the infimum $(λ_{min})$ and supremum $(λ_{max})$ of the Müntz sequence of real degrees of the integrands modelled by generalised polynomials. With an a-priori full analysis of the set of integrands to be integrated, the design and the action of such transformation ensures that the bounding monomials are integrated with machine-epsilon precision in double floating-point (f.p.) format [2] without resorting to less efficient schemes as adaptive or composite quadrature rules, singularity subtraction and cancellation methods with limited and uncontrolled precision. The effects of the action of the monomial transformation onto the original G-L samples are clearly visible by the clustering of the nodes around the endpoint singularity of the integrand function (see Fig. 1.2 of the user manual shipped with the source code and located in the MTQR/doc sub-directory as indicated in the root tree in Fig. 2.1 of said manual.)
Additional comments including restrictions and unusual features: To strictly secure the integration with finite-arithmetic precision at the order of the machine epsilon in double f.p. representation, higher-than-double data types are necessary to build the quadrature rule; furthermore to compute the optimal number of quadrature samples, the sole real root of a 7th degree polynomial [1, Equation (62)] has to be computed. For these reasons the proposed software depends on both the Boost's [3] Multiprecision header-only library and on the GNU Scientific Library, GSL [4]. Any source code using MTQR must be linked to those libraries which can be easily installed and configured for the system specific compiler in both Windows and Linux. Detailed information and assistance for successfully compiling and building applications with this library can be found in the aforementioned user manual.
MORe DWR: Space-time goal-oriented error control for incremental POD-based ROM for time-averaged goal functionals
2024, Journal of Computational Physics
In this work, the dual-weighted residual (DWR) method is applied to obtain an error-controlled incremental proper orthogonal decomposition (POD) based reduced order model. A novel approach called MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) is being introduced. It marries tensor-product space-time reduced-order modeling with time slabbing and an incremental POD basis generation with goal-oriented error control based on dual-weighted residual estimates. The error in the goal functional is being estimated during the simulation and the POD basis is being updated if the estimate exceeds a given threshold. This allows an adaptive enrichment of the POD basis in case of unforeseen changes in the solution behavior. Consequently, the offline phase can be skipped, the reduced-order model is being solved directly with the POD basis extracted from the solution on the first time slab and –if necessary– the POD basis is being enriched on-the-fly during the simulation with high-fidelity finite element solutions. Therefore, the full-order model solves can be reduced to a minimum, which is demonstrated on numerical tests for the heat equation and elastodynamics using time-averaged quantities of interest.
openFuelCell2: A new computational tool for fuel cells, electrolyzers, and other electrochemical devices and processes
2024, Computer Physics Communications
Fuel cells/electrolyzers are efficient and clean electrochemical devices that convert chemical energy directly into electricity and vice versa. They have attracted sustainable attention over the past decade from multiple experimental and numerical studies. However, detailed experimental investigations are typically expensive and challenging for providing a number of operating conditions and designs. Computational analysis offers an alternative approach for these studies. With the steadily increasing high-performance computing resources available, the limitations of numerical simulations have substantially decreased. This contribution details the design choice and code structure of modern electrochemical devices, which have been implemented as a versatile C++ library named openFuelCell2 within the open-source platform OpenFOAM, allowing for large-scale parallel calculations to be performed. The solver considers the major transport phenomena in a typical electrochemical device, including fluid flow, heat and mass transfer, species and charge transfer, and electrochemical reaction. This enables numerical simulations on popular electrochemical devices, such as fuel cells and electrolyzers, to be conducted. The paper also describes the domain decomposition, and parallel performance issues, as well as future applications.
Program Title: openFuelCell2
CPC Library link to program files: https://doi.org/10.17632/cvmb5xgy7h.1
Developer's repository link: https://github.com/openFuelCell2/openFuelCell2
Licensing provisions: GPLv3
Programming language: C++
Journal reference of previous version: S.B. Beale, H.-W. Choi, J.G. Pharoah, H.K. Roth, H. Jasak, D.H. Jeon, Open-source computational model of a solid oxide fuel cell, Computer Physics Communications 200 (2016) 15–26.
Does the new version supersede the previous version?: No
Reasons for new version: This new version aims to account for additional electrochemical applications.
Summary of revisions: The new version employs new code structure designs and is capable of considering two different electric potential fields and two-phase flows.
Nature of problem: This software library provides a set of models for simulating electrochemical devices, such as fuel cells and electrolyzers, as well as other similar devices. These systems consist of multiple components with distinct features but are coupled through properties such as temperature, species concentrations, and electric current/potentials. This software utilizes a computational fluid dynamics (CFD) approach to enable the simulation of multiregion and multiphysics problems, accounting for heat and mass transfer, single and two-phase flow, multicomponent diffusion, electron and ion transfer, and electrochemical reactions. This toolbox provides researchers with a powerful tool for studying electrochemical devices in a comprehensive and efficient manner.
Solution method: This software utilizes the finite volume method (FVM) to discretize and solve all conserved variables governed by partial differential equations. It employs a variant of the SIMPLE and PISO algorithms, known as PIMPLE, to solve the pressure-velocity coupling. All of the transport equations are solved sequentially, except for the enthalpy equation which is solved in an implicitly coupled manner within the global domain.
Additional comments including restrictions and unusual features:
- •
  The two-phase flow consists of continuous and dispersed phases, without resolving the phase interfaces.
- •
  The two-phase flow is solved with an Eulerian-Eulerian approach.
- •
  A local-time-stepping method was typically used in the two-phase solution.
- •
  The electrochemical reactions take place in volumetric regions.
- •
  Species diffusion is described by Fick's law.
Development and validation of a neutron transport solver with SP<inf>3</inf> method based on OpenFOAM
2024, Progress in Nuclear Energy
A deterministic neutron transport solver with ${SP}_{3}$ method based on OpenFOAM has been developed and validated in this paper. The physical models, numerical methods and schemes, boundary conditions, and solving strategy are discussed in detail. The approximate Marshak boundary condition of this solver is validated by the ISSA benchmark problem. Pin-by-pin lattice and simplified core neutron transport benchmark problems are introduced to make the validation of the solver. Numerical results show that ${SP}_{3}$ solver developed based on OpenFOAM has the promising prediction accuracy of $k_{e f f}$ , neutron scalar flux, fission power, and control rod worth. Parallel tests of ${SP}_{3}$ solver are also carried out under the million scale meshes, and the solver has an acceptable acceleration ratio and efficiency. No new computational methods or numerical schemes are proposed in this paper. Instead, the main new works in this paper are our detailed and quantified numerical validations of serval international neutron transport benchmark problems for ${SP}_{3}$ solver based on OpenFOAM. The capability of OpenFOAM in performing ${SP}_{3}$ calculations has been further tested and validated in this paper, which gives additional confidence that OpenFOAM can effectively be used for ${SP}_{3}$ analysis.
NekMesh: An open-source high-order mesh generation framework
2024, Computer Physics Communications
High-order spectral element simulations are now becoming increasingly popular within the computational modelling community, as they offer the potential to deliver increased accuracy at reduced cost compared to traditional low-order codes. However, to support accurate, high-fidelity simulations in complex industrial applications, there is a need to generate curvilinear meshes which robustly and accurately conform to geometrical features. This is, at present, a key challenge within the mesh generation community, with only a few open-source tools able to generate curvilinear meshes for complex geometries. We present NekMesh: an open-source mesh generation package which is designed to enable the generation of valid, high-quality curvilinear meshes of complex, three-dimensional geometries for performing high-order simulations. We outline the software architecture adopted in NekMesh, which uses a pipeline of processing modules to provide a flexible, CAD-independent high-order mesh processing tool, capable of both generating meshes for a wide range of use cases, as well as post-processing linear meshes from a range of input formats for use with high-order simulations. A number of examples in various application areas are presented, with a particular emphasis on challenging aeronautical and fluid dynamics test cases.
Program title: NekMesh (version 5.4.0)
CPC Library link to program files: https://doi.org/10.17632/d82hjm4v6r.1
Licensing provisions: MIT
Programming language: C++
External routines/libraries: Boost, TinyXML, OpenCASCADE, Triangle, TetGen, HDF5
Nature of problem: NekMesh is a high-order mesh generation framework with the goal of providing a robust framework to automate the process of generating valid meshes for complex three-dimensional CAD geometries.
Solution method: Energy minimisation, solid body models and other techniques based around high-order finite element methods.
Additional comments including restrictions and unusual features: The supplied cases require a system with 16384 MB of RAM for the automotive example and 32768 MB of RAM for the variational optimisation example.

View all citing articles on Scopus

^☆: This manuscript has been authored by UT-Battelle, LLC, USA under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).

¹: All authors of the paper are Principal Developers of the deal.II software library. They have all developed substantial pieces of deal.II, participate in patch review, and have participated in the writing of this article.

View full text

The deal.II finite element library: Design, features, and insights☆

Abstract

Introduction

Section snippets

Design considerations

Features

Lessons learned from the development of complex software

Education

Applications

Vision and future directions

Conclusions

Acknowledgments

Comput. Struct.

Comput. & Fluids

Adv. Eng. Softw.

Comput. Math. Appl.

SoftwareX

Appl. Math. Comput.

Comput. Phys. Comm.

Basic linear algebra subprograms for fortran usage

ACM Trans. Math. Software

LAPACK Users’ Guide

MPI: A Message-Passing Interface Standard Version 3.1

deal.II – a general purpose object oriented finite element library

ACM Trans. Math. Software

The deal.II library, version 8.4

J. Numer. Math.

The deal.II library, version 8.5

J. Numer. Math.

The deal.II library, version 9.0

J. Numer. Math.

The deal.II library, version 9.1

J. Numer. Math.

Adaptive multilevel methods with local smoothing for H1- and Hcurl-conforming high order finite element methods

SIAM J. Sci. Comput.

A flexible, parallel, adaptive geometric multigrid method for FEM

Algorithms and data structures for massively parallel generic adaptive finite element codes

ACM Trans. Math. Software

Data structures and requirements for hp finite element software

ACM Trans. Math. Software

Convergence study of the h-adaptive PUM and the hp-adaptive FEM applied to eigenvalue problems in quantum mechanics

Adv. Model. Simul. Eng. Sci.

Fast matrix-free evaluation of discontinuous Galerkin finite element operators

ACM Trans. Math. Software

Tools for the Solution of PDEs Defined on Curved Manifolds with Deal.II

Linearoperator benchmarks, version 1.0.0

Workstream – a design pattern for multicore-enabled finite element computations

ACM Trans. Math. Software

Using exact geometry information in finite element computations

A comprehensive set of tools for solving partial differential equations; diffpack

Libmesh: a C＋＋ library for parallel adaptive mesh refinement/coarsening simulations

Eng. Comput.

Session details: SPEC CPU2006 analysis

ACM SIGARCH Comput. Archit. News

Intel Threading Building Blocks: Outfitting C＋＋ for Multi-Core Processor Parallelism

An overview of the Trilinos project

ACM Trans. Math. Software

Trilinos web page

PETSc web page

PETSC Users Manual

P4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees

SIAM J. Sci. Comput.

Parallel tree algorithms for AMR and non-standard data access

Resolving the sign conflict problem for hp-hexahedral Nédélec elements with application to eddy current problems

Comp. Struct.

CuSPARSE

Adaptive multilevel methods with local smoothing for $H^{1}$ - and $H^{curl}$ -conforming high order finite element methods

Data structures and requirements for $h p$ finite element software