Mixed finite element method for a beam equation with the p-biharmonic operator

TL;DR

This paper develops a mixed finite element method for a nonlinear beam equation involving the p-biharmonic operator, establishing theoretical properties and validating through numerical simulations.

Contribution

It introduces a novel mixed finite element approach for the p-biharmonic beam equation, including existence, uniqueness, and convergence analysis.

Findings

Proved existence and uniqueness of weak solutions.

Established order of convergence for the method.

Validated theoretical results with Matlab simulations.

Abstract

In this paper, we consider a nonlinear beam equation with the p-biharmonic operator, where $1 < p < \infty$. Using a change of variable, we transform the problem into a system of differential equations and prove the existence, uniqueness and regularity of the weak solution by applying the Lax-Milgram theorem and classical results of functional analysis. We investigate the discrete formulation for that system and, with the aid of the Brouwer theorem, we show that the problem has a discrete solution. The uniqueness and stability of the discrete solution are obtained through classical methods. After establishing the order of convergence, we apply the mixed finite element method to obtain an algebraic system of equations. Finally, we implement the computational codes in Matlab software and perform the comparison between theory and simulations.