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An unfitted HDG method for a distributed optimal convection-diffusion control problem

Esteban Henríquez, Manuel Solano

Abstract

We analyze a high order unfitted hybridizable discontinuous Galerkin (HDG) method for an optimal control problem governed by a convection-diffusion equation posed in a domain with piecewise-wise $\mathcal{C}^2$ boundary $\partial Ω$. The computational domain $Ω_h$ does not necessarily fit $Ω$ and the Transfer Path Method (TPM) is used to transfer the boundary data from $\partial Ω$ to $\partial Ω_h$ through segments of direction $\boldsymbol{m}$. Under closeness conditions between $\partial Ω_h$ and $\partial Ω$ and on the transfer vector $\boldsymbol{m}$, we prove optimal order of convergence in the $L^2$-norm for all variables of the state and adjoint problems. We also show numerical examples to complement the theory.

An unfitted HDG method for a distributed optimal convection-diffusion control problem

Abstract

We analyze a high order unfitted hybridizable discontinuous Galerkin (HDG) method for an optimal control problem governed by a convection-diffusion equation posed in a domain with piecewise-wise boundary . The computational domain does not necessarily fit and the Transfer Path Method (TPM) is used to transfer the boundary data from to through segments of direction . Under closeness conditions between and and on the transfer vector , we prove optimal order of convergence in the -norm for all variables of the state and adjoint problems. We also show numerical examples to complement the theory.

Paper Structure

This paper contains 15 sections, 10 theorems, 94 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible triangulations.
  4. Spaces, mesh dependent inner-products and norms.
  5. Transferring paths and polynomial extrapolation.
  6. Closeness assumptions.
  7. $L^2$ and HDG projection.
  8. HDG formulation
  9. Error analysis
  10. Energy argument
  11. Duality argument
  12. Numerical experiments
  13. Conclusions
  14. Proofs of previous results
  15. Proof of Lemma \ref{['lem:Lambdas']}

Key Result

Lemma 1

If the aforementioned over $\bm m(\bm v_i)$ hold, then where $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Representation of a circle domain.
  • Figure 2: Representation of a kidney-shaped domain.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Corollary 1
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 9 more