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

Andrea Borio, Francesca Marcon, Maria Strazzullo

Abstract

In this work, we explore the application of the Virtual Element Methods for Neumann boundary Optimal Control Problems in saddle point formulation. The method is proposed for arbitrarily polynomial order of accuracy and general polygonal meshes. Our contribution includes a rigorous a priori error estimate that holds for general polynomial degree. On the numerical side, we present (i) an initial convergence test that reflects our theoretical findings, and (ii) a second test case based on a more application-oriented experiment. For the latter test, we focus on the role of VEM stabilization, conducting a detailed experimental analysis, and proposing an alternative structure-preserving strategy to circumvent issues related to the choice of the stabilization parameter.

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

Abstract

In this work, we explore the application of the Virtual Element Methods for Neumann boundary Optimal Control Problems in saddle point formulation. The method is proposed for arbitrarily polynomial order of accuracy and general polygonal meshes. Our contribution includes a rigorous a priori error estimate that holds for general polynomial degree. On the numerical side, we present (i) an initial convergence test that reflects our theoretical findings, and (ii) a second test case based on a more application-oriented experiment. For the latter test, we focus on the role of VEM stabilization, conducting a detailed experimental analysis, and proposing an alternative structure-preserving strategy to circumvent issues related to the choice of the stabilization parameter.

Paper Structure

This paper contains 9 sections, 8 theorems, 93 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. VEM Numerical Discretization
  4. Error Estimates
  5. Numerical Results
  6. Test 1
  7. Test 2 (sensitivity study on the stabilization parameter $\sigma$)
  8. Test 2 (structure preserving VEM)
  9. Conclusions

Key Result

Lemma 1

There exists $\eta_{\ast} \,, \eta^{\ast} \,, \mu^{\ast}>0$, independent of $h_E$, such that for each $w_h\in Y_{h}\ifstrempty{E}{}{(E)}$ and

Figures (12)

  • Figure 1: Test 1: schematic representation of the domain.
  • Figure 2: Test 1: meshes.
  • Figure 3: Test 1: convergence plots for the state variable $y$.
  • Figure 4: Test 1: convergence plots for the adjoint variable $p$.
  • Figure 5: Test 1: convergence plots for the control variable $u$.
  • ...and 7 more figures

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Proposition 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 4 more