[Paper Review] On the numerical approximation of $p$-Biharmonic and $\infty$-Biharmonic functions

Nikos Katzourakis, Tristan Pryer|arXiv (Cornell University)|Jan 25, 2017

Advanced Mathematical Modeling in Engineering3 references3 citations

TL;DR

This paper develops a C⁰ mixed finite element method for numerically approximating p-Biharmonic and ∞-Biharmonic functions, proving convergence of the discrete solution to the weak solution of the p-Biharmonic equation and rigorously passing to the limit p→∞ to capture ∞-Biharmonic solutions. The method reveals that ∞-Biharmonic functions are piecewise quadratic with complex interface patterns, even under simple boundary data.

ABSTRACT

In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in $L^{\infty}$. The associated equation, coined the $\infty$-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by $Δ^2_\infty u\, := (Δu)^3 | D (Δu) |^2 = 0.$ In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call $\infty$-Biharmonic functions. For fixed $p$ we design a mixed finite element scheme for the pre-limiting equation, the $p$-Bilaplacian $Δ^2_p u\, := Δ(| Δu |^{p-2} Δu) = 0.$ We prove convergence of the numerical solution to the weak solution of $Δ^2_p u = 0$ and show that we are able to pass to the limit $p o\infty$. We perform various tests aimed at understanding the nature of solutions of $Δ^2_\infty u$ and in 1-$d$ we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of $\mathcal D$-solutions.

Motivation & Objective

• To develop a stable and convergent numerical scheme for the p-Biharmonic equation, a fourth-order fully nonlinear PDE, under minimal regularity assumptions.
• To establish convergence of the mixed finite element method to the weak solution of the p-Biharmonic problem, with optimal convergence rates under additional regularity.
• To justify the p→∞ limit of the p-Biharmonic equation as the ∞-Biharmonic equation, a third-order fully nonlinear PDE not in divergence form.
• To numerically investigate the structure of ∞-Biharmonic functions, particularly their piecewise quadratic nature and complex interface patterns.
• To define and validate a weak solution concept—D-solutions—for the ∞-Biharmonic equation, given the lack of classical solutions.

Proposed method

• A mixed finite element formulation is constructed by introducing auxiliary variables for the Hessian and Laplacian, enabling the use of C⁰ finite elements instead of C¹ elements.
• The method employs a variational formulation based on minimizing a p-Biharmonic energy functional, rewritten in mixed form to decouple the fourth-order PDE.
• An inf-sup condition inspired by previous works ensures stability and enables convergence analysis with optimal rates under higher regularity.
• The discrete system is solved using a Galerkin approach with piecewise polynomial approximations, and convergence is proven via compactness and Young measure techniques.
• The p→∞ limit is analyzed using the theory of Calculus of Variations in L∞, leading to the ∞-Biharmonic equation as the formal limit of the p-Biharmonic PDE.
• Numerical experiments are conducted on 2D domains with varying m in boundary data to observe the emergence of piecewise constant Laplacians as p increases.

Experimental results

Research questions

• RQ1 How can a stable and convergent finite element method be designed for the p-Biharmonic equation without requiring C¹ finite elements?
• RQ2 What is the convergence behavior of the numerical solution to the p-Biharmonic equation, and can optimal convergence rates be established under regularity assumptions?
• RQ3 Can the limit p→∞ of the p-Biharmonic equation be numerically and analytically justified as the ∞-Biharmonic equation?
• RQ4 What structural properties do ∞-Biharmonic functions exhibit, particularly in terms of piecewise regularity and interface complexity?
• RQ5 How can a weak solution concept (D-solutions) be defined and numerically validated for the ∞-Biharmonic equation, which lacks classical solutions?

Key findings

• The mixed finite element method converges to the weak solution of the p-Biharmonic equation under minimal regularity assumptions, with convergence proven via compactness and Young measures.
• Under additional regularity, the method achieves optimal convergence rates that depend on the value of p, with higher p leading to improved stability and accuracy.
• As p increases, the numerical solution of the p-Biharmonic equation approaches the ∞-Biharmonic limit, with Δu tending toward piecewise constant functions, indicating the solution satisfies a Poisson equation with piecewise constant right-hand side.
• Numerical simulations for m=1,2,3 in the boundary data reveal that ∞-Biharmonic functions are piecewise quadratic over the domain, with highly complex and non-trivial interface patterns.
• The method successfully captures the D-solution concept for the ∞-Biharmonic equation, validating the weak formulation despite the absence of classical solutions.
• The observed convergence rates in numerical tests exceed theoretical predictions, suggesting potential for improvement via quasi-norm stabilization, consistent with known behavior in lower-order problems.