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A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem

Sijing Liu, Jinjin Zhang

Abstract

We consider a hybridizable discontinuous Galerkin (HDG) method for an elliptic distributed optimal control problem and we propose a balancing domain decomposition by constraints (BDDC) preconditioner to solve the discretized system. We establish an error estimate of the HDG methods with explicit tracking of a regularization parameter $β$. We observe that the BDDC preconditioner is robust with respect to $β$. Numerical results are shown to support our findings.

A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem

Abstract

We consider a hybridizable discontinuous Galerkin (HDG) method for an elliptic distributed optimal control problem and we propose a balancing domain decomposition by constraints (BDDC) preconditioner to solve the discretized system. We establish an error estimate of the HDG methods with explicit tracking of a regularization parameter . We observe that the BDDC preconditioner is robust with respect to . Numerical results are shown to support our findings.

Paper Structure

This paper contains 18 sections, 5 theorems, 78 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. HDG Discretization and Preliminary estimates
  3. HDG formulation
  4. Concise form
  5. Assumptions on the stabilizers $\tau_1$ and $\tau_2$
  6. Projection operators
  7. Convergence analysis for HDG methods
  8. Properties of $a_{1,h}$ and $a_{2,h}$
  9. An inf-sup condition
  10. Concrete error estimates
  11. A BDDC Preconditioner
  12. Operator form
  13. A characterization of the HDG Method
  14. Domain decomposition and a reduced subdomain interface problem
  15. A BDDC preconditioner
  16. ...and 3 more sections

Key Result

Lemma 3.1

Under Assumption assump:tau, we have, for any $((\bm{q}_h, y_h, \widehat{y}_h),(\bm{p}_h, p_h, \widehat{p}_h))\in (\bm{V}_h\times W_h\times\Lambda^0_h)\times(\bm{V}_h\times W_h\times\Lambda^0_h)$ and $((\bm{r}_1,w_1,\mu_1),(\bm{r}_2,w_2,\mu_2))\in (\bm{V}_h\times W_h\times\Lambda^0_h)\times(\bm{V}_h

Figures (6)

  • Figure 1: HDG degrees of freedom for $k=1$ and $k=2$
  • Figure 1: The degree of freedoms $\bm\lambda_{\Gamma}$ and $\bm\lambda_{I}$
  • Figure 1: Numerical solution with $\beta=10^{-4}$ and $h=1/96$.
  • Figure 2: The space $\widetilde{\bm{\Lambda}}_\Gamma$
  • Figure 3: Numerical solution with $\beta=10^{-6}$ and $h=1/96$.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Remark 1.1
  • Remark 2.1: Regularity
  • Remark 2.2
  • Lemma 3.1
  • Proof 1
  • Remark 3.2
  • Lemma 3.3
  • Proof 2
  • Lemma 3.4: Inf-sup
  • Proof 3
  • ...and 11 more