On the numerical approximation of <i>p</i>‐biharmonic and ∞‐biharmonic functions

Abstract

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given by urn:x-wiley:0749159X:media:num22295:num22295-math-0001 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 urn:x-wiley:0749159X:media:num22295:num22295-math-0002 We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞ . We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.