Finite Element Method Resolution and Error Estimators for p-Laplacian Problem with a New Boundary Condition

Abstract

In this article, we discuss the approach to solving a nonlinear PDE equation, specifically the p-Laplacian equation, with a general (nonlinear) boundary condition. We establish the existence and uniqueness of the solution, subject to certain assumptions outlined in this paper. To solve our nonlinear problem using the Finite Element Method (FEM), we derive an appropriate variational formulation. Additionally, we introduce a study of the residual a posteriori-error indicator, establishing both upper and lower bounds to control the error. The upper bound is determined using averaging interpolators in some quasi-norms defined by Barrett and Liu. Furthermore, we prove the equivalence between the residual error and the true error e = u − u_h. Lastly, we perform a simulation of the p-Laplacian problem in the L-shape domain using a Matlab program in two-dimensional space.