On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions. Issue 1 (21st November 2018)

• Record Type: Journal Article
• Authors: Katzourakis, Nikos; Pryer, Tristan
• Is Part Of: Numerical methods for partial differential equations. Volume 35:Issue 1(2019)
• Page Start: 155
• Page End: 180
• Publication Date: 2018-11-21
• Subjects: ∞‐Bilaplacian -- calculus of variations in L∞ -- finite element method -- fully nonlinear equations -- generalised solutions -- p‐Bilaplacian -- young measures; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22295
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : The ∞‐Bilaplacian is a third‐order fully nonlinear PDE given by 1 Δ ∞ 2 u ≔ Δ u 3 D Δ u 2 = 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 ‐Bilaplacian 2 Δ p 2 u ≔ Δ Δ u p − 2 Δ u = 0 . We prove convergence of the numerical solution to the weak solution of Δ 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 of Δ ∞ 2 u and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of D ‐solutions.