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

TL;DR

This work develops an ultra-weak three-field finite element formulation for the biharmonic equation, introducing ($u$,$σ$,$φ$) with $σ ≈ ∇u$ and discretising with Raviart-Thomas spaces to avoid $H^2$-conforming elements. It provides a rigorous saddle-point analysis establishing well-posedness and a priori error estimates, and extends the method to the time-dependent Extended Fisher-Kolmogorov equation with $g(u)=u^3-u$, including stability and convergence results. Numerical results in 2D and 3D validate optimal convergence for both simply supported and Cahn-Hilliard boundary conditions and reveal robustness with respect to the curvature parameter γ, including a 3D gear-domain example. Overall, the approach yields a stable, flexible mixed FEM for fourth-order problems on unstructured meshes, enabling efficient simulation of biharmonic and EFQ-type models.

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.

Figures (4)

• Figure 1: Approximate solutions for the Fisher--Kolmogorov problem at time $T=0.1$
• Figure 2: Approximate solutions for the biharmonic problem in 3D using simply supported boundary conditions. The results correspond to the lowest-order mixed scheme.
• Figure 3: Performance (error history and number of Newton iterations to converge) of the mixed finite element scheme for the extended Fisher--Kolmogorov problem, taking different values of the model parameter $\gamma$.
• Figure 4: Numerical solution of the extended Fisher--Kolmogorov equation on a gear model at $T=0.5$. Potential, gradient, and Lagrange multiplier obtained with the second-order scheme.

Theorems & Definitions (14)

• remark 1
• remark 2
• remark 3
• remark 4
• lemma 1
• proof
• lemma 2
• proof
• remark 5
• theorem 1