Very weak finite element methods: discretisation and applications

Abstract

Purpose

This study aims to propose and numerically assess different ways of discretising a very weak formulation of the Poisson problem.

Design/methodology/approach

We use integration by parts twice to shift smoothness requirements to the test functions, thereby allowing low-regularity data and solutions.

Findings

Various conforming discretisations are presented and tested, with numerical results indicating good accuracy and stability in different types of problems.

Originality/value

This is one of the first articles to propose and test concrete discretisations for very weak variational formulations in primal form. The numerical results, which include a problem based on real MRI data, indicate the potential of very weak finite element methods for tackling problems with low regularity.

Citation

Ramalho Queiroz Pacheco, D. (2024), "Very weak finite element methods: discretisation and applications", Engineering Computations, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/EC-10-2023-0699