Numerical analysis for an evolution equation with the p-biharmonic operator

Abstract

In this paper, we consider a parabolic equation with the p-biharmonic operator, where $p>1$. By employing a suitable change of variable, we transform the fourth-order nonlinear parabolic problem into a system of two second-order differential equations. We investigate the properties of the discretized solution in spatial and temporal variables. Using the Brouwer fixed point theorem we prove the existence of the discretized solution. Through classical functional analysis techniques we demonstrate the uniqueness and a priori estimates of the discretized solution. We establish the order of convergence in space and time, we establish the relationship between the temporal variable and the spatial variable, ensuring the existence of the convergence order. Additionally, we highlight that the change in variable carried out is extremely advantageous, as it allows us to obtain the order of convergence for the solution and its higher order derivatives using only lower-degree polynomials. Finally, using the finite element method with Lagrange basis, we implement the computational codes in Matlab software, considering the one and two-dimensional cases. We present three examples to illustrate and validate the theory.