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
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

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