Hybrid High-Order Methods on polyhedral Meshes – HHOMM

The goal of the project will be achieved through: (i) theoretical developments; (ii) applications to complex problems; (iii) development of computational methods and tools.

The results of this project have been published in top-ranking journals in Numerical Analysis and Scientific Computing.
They have also been disseminated through invited presentations in several prestigious international conferences.
Hybrid High-Order methods have also been taught in Ph.D. level courses in Europe.
A full list of publication as well as a selection of the most relevant dissemination events is available at the address www.math.univ-montp2.fr/~di-pietro/HHOMM.html.

The results of this project will foster relevant advances both in terms of fundamental tools and applications for Hybrid High-Order methods. They will also contribute significantly to the development of new generation polyhedral methods.

The full list of scientific publications issued from project HHOMM is available at the following address: www.math.univ-montp2.fr/~di-pietro/HHOMM.html

The numerical simulation of physical models based on Partial Differential Equations (PDEs) is an established tool in several domains.
Classical discretization methods, however, are often unable to provide the required flexibility in modern applications.
Their limitations include the type of computational meshes that can be handled, the possibility to modify the approximation order, the lack of robustness with respect to the variations of physical coefficients, etc.
Such limitations are particularly restrictive in application fields such as, e.g., nuclear waste management, where the computational mesh comes from seismic analysis and has to integrate complex geometrical features such as erosion, fractures and faults.

To answer these (and other) problems, a large effort has been devoted in recent years to the development of polyhedral structure-preserving methods.
This is nowadays a very competitive field of research where some of the most prominent groups in Numerical Analysis are active.
In the HHOMM project, we focus on a particularly promising instance of such methods, the Hybrid High-Order (HHO) schemes.
Relevant features of HHO schemes include: (i) the capability of handling general polyhedral meshes; (ii) dimension-independent construction; (iii) arbitrary approximation order; (iv) reproduction of desirable continuum properties at the discrete level; (vi) reduced computational cost.
Such features have generated significant interest, as testified by large-scale scientific initiatives (including the IHP quarter NMPDEs coordinated by the applicant) and industrial collaborations (with partners including Saint-Gobain, BRGM and EDF).

The goal of the HHOMM project is to help the HHO technology ripen and promote its use in engineering applications.
This will be achieved through:
(i) theoretical developments;
(ii) applications to complex problems;
(iii) development of computational methods and tools.

Pursuing theoretical developments will allow both to meet practical needs (treatment of curved faces, adaptivity) and to improve knowledge in the field.
Considering applications to complex problems will, on the other hand, be a crucial step to promote HHO methods in the engineering community. It will also provide the opportunity to foster and develop (possibly multi-disciplinary) industrial and academic collaborations. The focus will be here on problems of potential industrial interest in fluid- and solid-mechanics and electromagnetism.
Finally, developing practical tools is a necessary step to develop a reference implementation paradigm, a key step to compete with classical methods such as Finite Elements.
The practical demonstration will be based on the "hho" software platform (formerly known as "polyC++"), which has recently been registered by the applicant as a shared property of University of Montpellier and CNRS.

To achieve the above goals, the HHOMM project relies on a well-oiled team composed of internationally renowned scientists with a strong background in Numerical Analysis and Scientific Computing, as well as a consolidated experience in the proposed application fields.
The permanent researchers will be joined by a post-doctoral researcher and a PhD student.
The post-doctoral researcher will work on fundamental developments and their demonstration using the "hho" platform.
The PhD student will develop the application of HHO methods to electromagnetism.

The results obtained within this project will make the object of publication in top-ranking international journals in Numerical Analysis and Scientific Computing.
Two major events will allow the dissemination of the results: one conference that will take place in Paris within the IHP quarter NMPDEs and one workshop that will allow to draw conclusions towards the end of the project.
One key project deliverable will be the "hho" software platform, which we will make available to the scientific computing community by the end of the HHOMM project.