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.

中文翻译：

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非常弱的有限元方法：离散化和应用

目的

本研究旨在提出并数值评估离散泊松问题的非常弱的表述的不同方法。

设计/方法论/途径

我们两次使用分部积分将平滑度要求转移到测试函数，从而允许低规律性数据和解决方案。

发现

提出并测试了各种一致的离散化，数值结果表明在不同类型的问题中具有良好的准确性和稳定性。

原创性/价值

这是最早提出并测试原始形式的非常弱变分公式的具体离散化的文章之一。数值结果（包括基于真实 MRI 数据的问题）表明，非常弱的有限元方法在解决低规律性问题方面具有潜力。