Stabilization-Free General Order Virtual Element Methods for Neumann Boundary Optimal Control Problems in Saddle Point Formulation

TL;DR

This paper introduces a stabilization-free virtual element method for Neumann boundary optimal control problems, providing error estimates and numerical tests demonstrating its effectiveness across arbitrary polynomial orders and polygonal meshes.

Contribution

It develops a stabilization-free VEM approach for saddle point problems in optimal control, with rigorous error analysis and diverse numerical experiments.

Findings

The method achieves optimal convergence rates.

Stabilization-free approach reduces parameter tuning issues.

Numerical tests confirm theoretical error estimates.

Abstract

In this work, we explore the application of Stabilization-Free Virtual Element Methods for Neumann boundary Optimal Control Problems in saddle point formulation. The method is proposed for arbitrary polynomial order of accuracy and general polygonal meshes. Our contribution includes a rigorous a priori error estimate that holds for general polynomial order. On the numerical side, we present (i) an initial convergence test that reflects our theoretical findings, (ii) a second test analyzing the role of the stabilization term in the Virtual Element Method (VEM) formulation and its influence on the approximation error, and (iii) a third test case based on a more application-oriented experiment. The stabilization-free approach is proposed as an alternative strategy to circumvent issues related to the choice of the stabilization parameter in standard VEM formulations.