Robust Multigrid Methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem

Sijing Liu

doi:10.1515/cmam-2023-0132

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Robust Multigrid Methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem

Sijing Liu

Published/Copyright: January 30, 2024

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From the journal Computational Methods in Applied Mathematics Volume 25 Issue 1

Abstract

We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem, and we propose multigrid methods to solve the discretized system. We prove that the 𝑊-cycle algorithm is uniformly convergent in the energy norm and is robust with respect to a regularization parameter on convex domains. Numerical results are shown for both 𝑊-cycle and 𝑉-cycle algorithms.

Keywords: Elliptic Distributed Optimal Control Problems; General State Equations; Multigrid Methods; Discontinuous Galerkin Methods

MSC 2020: 49J20; 49M41; 65N30; 65N55

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-1929284

Funding statement: The revision of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1929284 while the author was in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Numerical PDEs: Analysis, Algorithms, and Data Challenges” program.

A Proofs of (3.9) and (3.10)

For T ∈ T h and v ∈ H 1 + s ⁢ ( Ω ) , where s ∈ ( 1 2 , 1 ] , the following trace inequalities with scaling is standard (cf. [25, Lemma 7.2] and [21, Proposition 3.1]):

(A.1) ∥ v ∥ L 2 ⁢ ( ∂ T ) ≤ C ⁢ ( h T − 1 2 ⁢ ∥ v ∥ L 2 ⁢ ( T ) + h T s − 1 2 ⁢ | v | H s ⁢ ( T ) ) .

The following discrete Poincaŕe inequality for DG functions [8, 4, 20] is valid for all v ∈ V h :

(A.2) ∥ v ∥ L 2 ⁢ ( Ω ) 2 ≤ C ⁢ ( ∑ T ∈ Ω ∥ ∇ v ∥ L 2 ⁢ ( T ) 2 + ∑ e ∈ ∂ T 1 h e ⁢ ∥ [ v ] ∥ L 2 ⁢ ( e ) 2 ) .

Proof

It is well known that [3, 37, 15]

a h sip ⁢ ( w , v ) ≤ C ⁢ ∥ w ∥ 1 , h ⁢ ∥ v ∥ 1 , h for all ⁢ w , v ∈ V + V h , a h sip ⁢ ( v , v ) ≥ C ⁢ ∥ v ∥ 1 , h 2 for all ⁢ v ∈ V h .

For the advection-reaction term, we have, for all w , v ∈ V + V h ,

a h ar ⁢ ( w , v ) = ∑ T ∈ T h ( ζ ⋅ ∇ w + γ ⁢ w , v ) T − ∑ e ∈ E h i ∪ E h b , − ( n ⋅ ζ ⁢ [ w ] , { v } ) e ≲ ( ∑ T ∈ T h ∥ ∇ w ∥ L 2 ⁢ ( T ) 2 ) 1 2 ⁢ ∥ v ∥ L 2 ⁢ ( Ω ) + ∥ w ∥ L 2 ⁢ ( Ω ) ⁢ ∥ v ∥ L 2 ⁢ ( Ω ) + ( ∑ e ∈ E h i ∪ E h b , − σ h e ⁢ ∥ [ w ] ∥ L 2 ⁢ ( e ) 2 ) 1 2 ⁢ ( ∑ e ∈ E h i ∪ E h b , − h e σ ⁢ ∥ { v } ∥ L 2 ⁢ ( e ) 2 ) 1 2 ≲ | | | w | | | h ⁢ | | | v | | | h ,

where we use ζ ∈ [ W 1 , ∞ ⁢ ( Ω ) ] 2 , γ ∈ L ∞ ⁢ ( Ω ) , (A.1) and (A.2). Furthermore, upon integration by parts, we have, for all v ∈ V h ,

a h ar ⁢ ( v , v ) = ∑ T ∈ T h ( ζ ⋅ ∇ v + γ ⁢ v , v ) T − ∑ e ∈ E h i ∪ E h b , − ( n ⋅ ζ ⁢ [ v ] , { v } ) e = ∑ T ∈ T h ( ( γ − 1 2 ⁢ ∇ ⋅ ζ ) ⁢ v , v ) T + ∑ T ∈ T h ∫ ∂ T 1 2 ⁢ ( ζ ⋅ n ) ⁢ v 2 ⁢ d s − ∑ e ∈ E h i ∪ E h b , − ∫ e ζ ⋅ n ⁢ [ v ] ⁢ { v } ⁢ d s = ∑ T ∈ T h ( ( γ − 1 2 ⁢ ∇ ⋅ ζ ) ⁢ v , v ) T + ∫ ∂ Ω 1 2 ⁢ | ζ ⋅ n | ⁢ v 2 ⁢ d s .

By assumption (1.4), we immediately have a h ar ⁢ ( v , v ) ≥ 0 . This finishes the proof. ∎

B A Proof of (3.22)

Proof

It follows from (3.20) that

(B.1) ∥ p − p h ∥ L 2 ⁢ ( Ω ) 2 + ∥ y − y h ∥ L 2 ⁢ ( Ω ) 2 = ( β 1 2 ⁢ ( − Δ ⁢ ξ + ζ ⋅ ∇ ξ + γ ⁢ ξ ) − θ , p − p h ) L 2 ⁢ ( Ω ) + ( − ξ + β 1 2 ⁢ ( Δ ⁢ θ + ζ ⋅ ∇ θ − ( γ − ∇ ⋅ ζ ) ⁢ θ ) , y − y h ) L 2 ⁢ ( Ω ) = β 1 2 ⁢ ( − Δ ⁢ ξ , p − p h ) L 2 ⁢ ( Ω ) + β 1 2 ⁢ ∑ T ∈ T h ( ζ ⋅ ∇ ξ + γ ⁢ ξ , p − p h ) T − ( θ , p − p h ) L 2 ⁢ ( Ω ) − ( ξ , y − y h ) L 2 ⁢ ( Ω ) + β 1 2 ⁢ ( Δ ⁢ θ , y − y h ) L 2 ⁢ ( Ω ) + β 1 2 ⁢ ∑ T ∈ T h ( ζ ⋅ ∇ θ − ( γ − ∇ ⋅ ζ ) ⁢ θ , y − y h ) T .

By the consistency of the SIP method (cf. [37, 3]), we have

(B.2) ( − Δ ⁢ ξ , p − p h ) = a h sip ⁢ ( ξ , p − p h ) and ( Δ ⁢ θ , y − y h ) = − a h sip ⁢ ( y − y h , θ ) .

For the last term in (B.1), it follows from integration by parts that

(B.3) ∑ T ∈ T h ( ζ ⋅ ∇ θ − ( γ − ∇ ⋅ ζ ) ⁢ θ , y − y h ) T = ∑ T ∈ T h ( − ζ ⋅ ∇ ( y − y h ) , θ ) T − ( γ ⁢ ( y − y h ) , θ ) T + ∑ T ∈ T h ∫ ∂ T ( ζ ⋅ n ) ⁢ ( y − y h ) ⁢ θ ⁢ d s .

The last term in (B.3) can be rewritten as the following [3, 24]:

(B.4) ∑ T ∈ T h ∫ ∂ T ( ζ ⋅ n ) ⁢ ( y − y h ) ⁢ θ ⁢ d s = ∑ e ∈ E h i ∫ e ζ ⋅ n ⁢ [ ( y − y h ) ⁢ θ ] ⁢ d s + ∑ e ∈ E h b ∫ e ζ ⋅ n ⁢ ( y − y h ) ⁢ θ ⁢ d s = ∑ e ∈ E h i ∫ e ζ ⋅ n ⁢ [ y − y h ] ⁢ { θ } ⁢ d s + ∑ e ∈ E h i ∫ e ζ ⋅ n ⁢ { y − y h } ⁢ [ θ ] ⁢ d s + ∑ e ∈ E h b ∫ e ζ ⋅ n ⁢ ( y − y h ) ⁢ θ ⁢ d s .

It then follows from [ θ ] = 0 on interior edges and θ = 0 on ∂ Ω that

(B.5) ∑ T ∈ T h ∫ ∂ T ( ζ ⋅ n ) ⁢ ( y − y h ) ⁢ θ ⁢ d s = ∑ e ∈ E h i ∪ E h b , − ∫ e ζ ⋅ n ⁢ [ y − y h ] ⁢ { θ } ⁢ d s .

According to (B.3)–(B.5), we conclude

(B.6) ∑ T ∈ T h ( ζ ⋅ ∇ θ − ( γ − ∇ ⋅ ζ ) ⁢ θ , y − y h ) T = − a h ar ⁢ ( y − y h , θ ) .

Similarly, we can show

(B.7) ∑ T ∈ T h ( ζ ⋅ ∇ ξ + γ ⁢ ξ , p − p h ) T = a h ar ⁢ ( ξ , p − p h ) .

Therefore, we obtain the following by (B.2), (B.6), (B.7), (3.3) and (3.2):

∥ p − p h ∥ L 2 ⁢ ( Ω ) 2 + ∥ y − y h ∥ L 2 ⁢ ( Ω ) 2 = β 1 2 ⁢ a h ⁢ ( ξ , p − p h ) − ( θ , p − p h ) L 2 ⁢ ( Ω ) − ( ξ , y − y h ) L 2 ⁢ ( Ω ) − β 1 2 ⁢ a h ⁢ ( y − y h , θ ) = B h ⁢ ( ( p − p h , y − y h ) , ( ξ , θ ) ) . ∎

Acknowledgements

The author would like to thank Prof. Susanne C. Brenner, Prof. Li-Yeng Sung and Prof. Yi Zhang for the helpful discussion regarding this project.

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Received: 2023-05-16

Revised: 2024-01-12

Accepted: 2024-01-17

Published Online: 2024-01-30

Published in Print: 2025-01-01

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Articles in the same Issue

https://doi.org/10.1515/cmam-2023-0132

Keywords for this article

Elliptic Distributed Optimal Control Problems; General State Equations; Multigrid Methods; Discontinuous Galerkin Methods