On the numerical approximation of p ‐biharmonic and ∞ ‐biharmonic functions

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given byΔ ∞ 2 u ≔Δ u 3D Δ u2 = 0 .In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞ ‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p ‐BilaplacianΔ p 2 u ≔ ΔΔ up − 2 Δ u = 0 .We prove convergence of the numerical solution to the weak solution ofΔ p 2 u = 0 and show that we are able to pass to the limit p  → ∞ . We perform various tests aimed at understanding the nature of solutions ofΔ ∞ 2 u and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of D ‐solutions.