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On the numerical approximation of p ‐biharmonic and ∞ ‐biharmonic functions
Author(s) -
Katzourakis Nikos,
Pryer Tristan
Publication year - 2019
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22295
Subject(s) - biharmonic equation , mathematics , discretization , convergence (economics) , limit (mathematics) , finite element method , scheme (mathematics) , mathematical analysis , nonlinear system , order (exchange) , boundary value problem , physics , finance , quantum mechanics , economics , thermodynamics , economic growth
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.