Introduction

About Hermes2D

Hermes2D is a free C++/Python library for rapid development of adaptive hp-FEM and hp-DG solvers for partial differential equations (PDE) and multiphysics PDE systems. The developer team includes the hp-FEM group at the University of Nevada, Reno and their collaborators from numerous places around the globe.

The standard way to use Hermes2D is to write short C++ user programs, but for those who prefer to use a graphical interface, there is an interactive GUI Agros2D. We also provide an interactive online lab where the user can compute with Hermes2D and other FEM codes in FEMhub via any web browser without having to install anything (CPU time is on us).

Although Hermes2D is much younger than major FEM packages, it is loaded with unique technology and its user base is growing fast. We hope that you will enjoy the software and that you will find this documentation useful. Let us know if you find mistakes or with any improvement suggestions. Anyone who contributes a patch becomes automatically a co-author of the code.

The library is available under the GPL license (Version 2, 1991). In the following, we will abbreviate Hermes2D with Hermes.

About this Document

Prior to reading this document, we recommend that you install Hermes using instructions on its home page, and subscribe to the mailing list. Our mailing list is a very active place where you will get any questions answered quickly. You can also follow the development via the group activity list that contains a weekly log of all core team members.

The best way of reading this tutorial is to run the code at the same time. After making your way through the tutorial, you may want to browse the directories with benchmarks and examples that contain a variety of different PDE models. If you create an interesting model using Hermes, let us know and we will add it to the repository.

The source code can be viewed in the git repository, and all tutorial examples can be found in the directory tutorial/. For the 1D and 3D codes, see the Hermes1D and Hermes3D home pages, respectively.

User and Developer Documentation

User documentation (tutorial, benchmarks, examples) can be found in the directory ‘doc/’. Type ‘make html’ there to build it. The documentation is available online at http://hpfem.org/hermes2d/doc/index.html.

To compile the C++ reference manual, go to ‘hermes2d/doc.cpp/’. There type ‘doxygen hermes2d.lib-real.doxyfile’ to build references for the real version, or ‘doxygen hermes2d.lib-cplx.doxyfile’ to build refs for the complex version. The html files are in ‘h2d-real/html/index.html’ and ‘h2d-real/cplx/index.html’, respectively. This documentation is also available online at http://hpfem.org/hermes2d/doc.cpp/h2d-real/html/index.html and http://hpfem.org/hermes2d/doc.cpp/h2d-cplx/html/index.html, respectively.

Mathematical Background

Main strengths of Hermes are

• adaptive hp-FEM and hp-DG methods,
• adaptivity for time-dependent problems on dynamically-changing hp-meshes, and
• monolithic discretization of multiphysics problems via multimesh hp-FEM/hp-DG.

The following list describes the above in more detail:

• Mature hp-adaptivity algorithms. Hermes puts a major emphasis on error control and automatic adaptivity. Practitioners know well how painful it is to use automatic adaptivity in conjunction with standard lower-order approximations such as linear or quadratic elements - the error decreases somehow during a few initial adaptivity steps, but then it slows down and it does not help to invest more unknowns or CPU time. This is typical for low-order methods. In contrast to this, the exponentially-convergent adaptive hp-FEM and hp-DG do not have this problem - the error drops steadily and fast during adaptivity all the way to the desired accuracy. The following graph shows typical convergence rates of h-FEM with linear elements, h-FEM with quadratic elements, and hp-FEM on a log-log scale:

Same graphs as above but now in terms of CPU time:

• Wide applicability. Hermes is PDE-independent. Standard FEM codes are designed to solve some narrow class(es) of PDE problems (such as elliptic equations, fluid dynamics, electromagnetics etc.). Hermes does not employ any technique or algorithm that would limit its applicability to some particular class(es) of PDE problems. Automatic adaptivity is guided by a universal computational a-posteriori error estimate that works in the same way for any PDE. Of course this does not mean that the algorithms perform equally well on all PDE - some equations simply are more difficult to solve than others. However, Hermes allows you to tackle an arbitrary PDE or multiphysics PDE system and add your own equation-specific extensions if necessary. Visit the hp-FEM group home page and especially the gallery to see numerous examples.

• Arbitrary-level hanging nodes. Hermes has a unique original methodology for handling irregular meshes with arbitrary-level hanging nodes. This means that extremely small elements can be adjacent to very large ones. When an element is refined, its neighbors are never split forcefully as in conventional adaptivity algorithms. It is well known that approximations with one-level hanging nodes are more efficient compared to regular meshes. However, the technique of arbitrary-level hanging nodes brings this to a perfection.

• Multimesh hp-FEM. Various physical fields or solution components in multiphysics problems can be approximated on individual meshes, combining quality , , , and conforming higher-order elements. Due to a unique original methodology, no error is caused by operator splitting, transferring data between different meshes, and the like. The following figure illustrates a coupled problem of heat and moisture transfer in massive concrete walls of a nuclear reactor vessel.

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• Dynamical meshes for time-dependent problems. In time-dependent problems, different physical fields or solution components can be approximated on individual meshes that evolve in time independently of each other. Due to a unique original methodology, no error is caused by transfering solution data between different meshes and time levels. No such transfer takes place in the multimesh hp-FEM - the discretization of the time-dependent PDE system is monolithic.

Interactive Web Accessibility

• Interactive web usage. You can use Hermes (and other major open source FEM codes) remotely via any web browser, using the FEMhub Online Numerical Methods Laboratory. Your hardware will not be used as the online lab is powered by the University of Nevada, Reno (UNR) high-performance computing facility (Research Grid). In this way you can compute with Hermes using any platform that supports web browsing, such as an iPhone:

See the Hermes home page for more information. An overview of books, journal articles, conference proceedings papers and talks about Hermes and adaptive hp-FEM can be found in its publications section.

Citing Hermes

If you use Hermes for your work, please be so kind to include some of the references below as appropriate.

Monographs:

@Book{Hermes-book1,
   author = {P. Solin, K. Segeth, I. Dolezel},
   title = {Higher-Order Finite Element Methods},
   publisher = {Chapman & Hall / CRC Press},
   year = {2003}
}

@Book{Hermes-book2,
   author = {P. Solin},
   title = {Partial Differential Equations and the Finite Element Method},
   publisher = {J. Wiley & Sons},
   year = {2005}
}

Reference to the Hermes open-source project:

@Manual{Hermes-project,
   title =  {Hermes - Higher-Order Modular Finite Element System (User's Guide)},
   author = {P. Solin et al.},
   url =    {http://hpfem.org/}
}

Underlying algorithms (hanging nodes, adaptivity, shape functions):

@Article{Hermes-hanging-nodes,
  author = {P. Solin, J. Cerveny, I. Dolezel},
  title = {Arbitrary-Level Hanging Nodes and Automatic Adaptivity in the hp-FEM},
  journal = {Math. Comput. Simul.},
  year = {2008},
  volume = {77},
  pages = {117 - 132}
}

@Article{Hermes-adaptivity,
  author = {P. Solin, D. Andrs, J. Cerveny, M. Simko},
  title = {PDE-Independent Adaptive hp-FEM Based on Hierarchic Extension of Finite Element Spaces},
  journal = {J. Comput. Appl. Math.},
  year = {2010},
  volume = {233},
  pages = {3086-3094}
}

@Article{Hermes-shape-functions,
  author = {P. Solin, T. Vejchodsky},
  title = {Higher-Order Finite Elements Based on Generalized Eigenfunctions of the Laplacian},
  journal = {Int. J. Numer. Methods Engrg},
  year = {2007},
  volume = {73},
  pages = {1374 - 1394}
}

Topical papers from various application areas:

@Article{Hermes-multiphysics,
  author = {P. Solin, L. Dubcova, J. Kruis},
  title = {Adaptive hp-FEM with Dynamical Meshes for Transient Heat and Moisture Transfer Problems},
  journal = {J. Comput. Appl. Math},
  year = {2010},
  volume = {233},
  pages = {3103-3112}
}

@Article{Hermes-solid-mechanics,
  author = {P. Solin, J. Cerveny, L. Dubcova, D. Andrs},
  title = {Monolithic Discretization of Linear Thermoelasticity Problems via Adaptive Multimesh hp-FEM},
  journal = {J. Comput. Appl. Math},
  status = {published online},
  doi = {doi 10.1016/j.cam.2009.08.092},
  year = {2009}
}

@Article{Hermes-electromagnetics,
  author = {L. Dubcova, P. Solin, J. Cerveny, P. Kus},
  title = {Space and Time Adaptive Two-Mesh hp-FEM for Transient Microwave Heating Problems},
  journal = {Electromagnetics},
  year = {2010},
  volume = {30},
  pages = {23 - 40}
}

@Article{Hermes-fluid-mechanics,
  author = {P. Solin, J. Cerveny, L. Dubcova, I. Dolezel},
  title = {Multi-Mesh hp-FEM for Thermally Conductive Incompressible Flow},
  journal = {Proceedings of ECCOMAS Conference COUPLED PROBLEMS 2007 (M. Papadrakakis, E. Onate,
             B. Schrefler Eds.), CIMNE, Barcelona},
  year = {2007},
  pages = {677 - 680}
}

Other papers that may be still closer to what you need can be found in the publications section of the hp-FEM group home page or on Pavel Solin’s home page.