Localized orthogonal decomposition for high-order, hybrid finite elements

Duration: 03/2024 – 02/2026
Funding scheme: Funding for Mobility Cooperation with Germany
Funder: Research Council of Finland
Budget: 18,000 €
My role: Finnish principal investigator (transferred to Zhi-Song Liu in 09/2024)
Sibling project: High-order hybrid multiscale methods for rough heterogeneous structures
German principal investigator: Roland Maier

Public description

Many natural processes involve multiple scales, such as fluid flow in porous materials or wave propagation through layered soil. These systems have microscopic scales, describing material properties and textures, and macroscopic scales, where observable phenomena occur. The same principle applies in manufacturing, for example, when producing fiber-reinforced composites that alter material behavior.

Mathematically, partial differential equations with coefficients varying on a microscopic scale describe such processes. While typically only practical macroscopic information is of interest, discarding microscopic features in numerical simulations often fails to provide the desired results. However, fully resolving microscopic coefficients is computationally prohibitive.

Numerical homogenization methods offer a possible solution to this problem. This project aims to improve their performance by combining state-of-the-art hybrid finite elements with high-order multiscale approaches.

Scientific abstract

This is achieved by solving a set of auxiliary local problems in an offline phase, which then significantly lower computation times to solve the problem of interest in an online stage. Such an approach is particularly beneficial if similar problems have to be solved multiple times, as in the context of time-dependent PDEs and uncertainty quantification.

Current research aims to further improve the locality of the auxiliary problems while enhancing the accuracy, for instance, with higher-order approximations. In this project, we aim to combine the efficient multiscale method known as localized orthogonal decomposition with the advantages of hybrid finite element schemes. Doing so allows us to

  • exploit the benefits (symmetric positive definite matrices, super-convergence, etc.) of hybrid discontinuous Galerkin and hybrid high-order methods,
  • extend the construction and allow for higher-order convergence rates without restrictive regularity assumptions,
  • investigate super-localization properties.

Keywords

high-order approximation, hybrid discontinuous Galerkin, hybrid high-order method, localized orthogonal decomposition, multiscale problems, rough coefficients, super-localized orthogonal decomposition

Related publications

Kaarnioja, Vesa; Rupp, Andreas

Quasi-Monte Carlo and discontinuous Galerkin Journal Article

In: Electronic Transactions on Numerical Analysis, vol. 60, pp. 589–617, 2024, ISSN: 1068-9613.

Abstract | Links | BibTeX

Altunay, Rabia; Vesterinen, Kalevi; Alander, Pasi; Immonen, Eero; Rupp, Andreas; Roininen, Lassi

Denture reinforcement via topology optimization Journal Article Forthcoming

In: Medical Engineering & Physics, Forthcoming, ISSN: 1350-4533.

Abstract | Links | BibTeX

Liu, Zhi-Song; Maier, Roland; Rupp, Andreas

Numerical homogenization by continuous super-resolution Online Forthcoming

Forthcoming, visited: 13.11.2024.

Abstract | Links | BibTeX

Liu, Zhi-Song; Büttner, Markus; Aizinger, Vadym; Rupp, Andreas

Super-resolution works for coastal simulations Online Forthcoming

Forthcoming, visited: 30.08.2024.

Abstract | Links | BibTeX

Hauck, Moritz; Målqvist, Axel; Rupp, Andreas

Arbitrary order approximations at constant cost for Timoshenko beam network models Online Forthcoming

Forthcoming, visited: 19.07.2024.

Abstract | Links | BibTeX

Vedral, Joshua; Rupp, Andreas; Kuzmin, Dmitri

Strongly consistent low-dissipation WENO schemes for finite elements Online Forthcoming

Forthcoming, visited: 08.07.2024.

Abstract | Links | BibTeX

Cheng, Hanz Martin; Helin, Tapio; Manninen, Ville-Petteri; Holopainen, Timo; Jokinen, Juha; Sorvari, Samu; Rupp, Andreas

Recovery of transversely-isotropic elastic material parameters in induction motor rotors Online Forthcoming

Forthcoming, visited: 10.05.2024.

Abstract | Links | BibTeX

Di Pietro, Daniele Antonio; Dong, Zhaonan; Kanschat, Guido; Matalon, Pierre; Rupp, Andreas

Homogeneous multigrid for hybrid discretizations: application to HHO methods Online Forthcoming

Forthcoming, visited: 25.03.2024.

Abstract | Links | BibTeX

Lu, Peipei; Maier, Roland; Rupp, Andreas

A localized orthogonal decomposition strategy for hybrid discontinuous Galerkin methods Online Forthcoming

Forthcoming, visited: 28.07.2023.

Abstract | Links | BibTeX

Kazarnikov, Alexey; Ray, Nadja; Haario, Heikki; Lappalainen, Joona; Rupp, Andreas

Parameter estimation for cellular automata Online Forthcoming

Forthcoming, visited: 01.02.2023.

Abstract | Links | BibTeX