A Hybrid High-Order Method for a Class of Strongly Nonlinear Elliptic Boundary Value Problems

TL;DR

This paper introduces a Hybrid High-Order finite element method for solving strongly nonlinear elliptic boundary value problems, providing theoretical analysis and numerical validation of its accuracy and convergence.

Contribution

The paper develops a novel HHO discretization for nonlinear elliptic problems, including stability analysis, existence proof, and optimal error estimates.

Findings

Proved well-posedness of the HHO scheme.

Established optimal a priori error estimates.

Numerical experiments confirm convergence and accuracy.

Abstract

In this article, we design and analyze a Hybrid High-Order (HHO) finite element approximation for a class of strongly nonlinear boundary value problems. We consider an HHO discretization for a suitable linearized problem and show its well-posedness using the Gardings type inequality. The essential ingredients for the HHO approximation involve local reconstruction and high-order stabilization. We establish the existence of a unique solution for the HHO approximation using the Brouwer fixed point theorem and contraction principle. We derive an optimal order a priori error estimate in the discrete energy norm. Numerical experiments are performed to illustrate the convergence histories.