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

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.

💡 Research Summary

The paper introduces an ultra‑weak three‑field mixed finite‑element formulation for fourth‑order partial differential equations, focusing on the biharmonic problem and the time‑dependent extended Fisher–Kolmogorov (EFK) equation. The three unknowns are the scalar solution (u), its gradient (\boldsymbol\sigma=\nabla u), and a Lagrange multiplier (\phi) that enforces the gradient constraint. By integrating the constraint by parts, the authors obtain a formulation in which (u) belongs only to (L^{2}(\Omega)); this is why the method is called “ultra‑weak”. The auxiliary variables (\boldsymbol\sigma) and (\phi) are discretised with Raviart–Thomas (RT) finite‑element spaces of order (k), while (u) is approximated by standard discontinuous piecewise polynomials of the same degree.

The continuous problem is written as a saddle‑point system \