Very weak finite element methods: discretisation and applications

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.