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arXiv · 2601.22587

An ultra-weak three-field finite element formulation for the biharmonic and extended Fisher--Kolmogorov equations

Rekha Khot·Bishnu P. Lamichhane·Ricardo Ruiz-Baier
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Abstract

This paper discusses a so-called ultra-weak three-field formulation of the biharmonic problem where the solution, its gradient, and an additional Lagrange multiplier are the three unknowns. We establish the well-posedness of the problem using the abstract theory for saddle-point problems, and develop a conforming finite element scheme based on Raviart--Thomas discretisations of the two auxiliary variables. The well-posedness of the discrete formulation and the corresponding a priori error estimate are proved using a discrete inf-sup condition. We further extend the analysis to the time-dependent semilinear equation, namely extended Fisher--Kolmogorov equation. We present a few numerical examples to demonstrate the performance of our approach.

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Rekha Khot, Bishnu P. Lamichhane, Ricardo Ruiz-Baier. 2026-01-30. An ultra-weak three-field finite element formulation for the biharmonic and extended Fisher--Kolmogorov equations. https://arxiv.org/abs/2601.22587

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