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A 𝑃1 Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints

  • Susanne C. Brenner EMAIL logo , Sijing Liu and Li-Yeng Sung
Published/Copyright: June 23, 2021
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Computational Methods in Applied Mathematics
From the journal Computational Methods in Applied Mathematics Volume 21 Issue 4

Abstract

We investigate a P1 finite element method for an elliptic distributed optimal control problem with pointwise state constraints and a state equation that includes advective/convective and reactive terms. The convergence of this method can be established for general polygonal/polyhedral domains that are not necessarily convex. The discrete problem is a strictly convex quadratic program with box constraints that can be solved efficiently by a primal-dual active set algorithm.

Keywords: Elliptic Distributed Optimal Control Problems; General State Equation; Pointwise State Constraints
MSC 2010: 65N30; 65K15; 90C20

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-19-13035

Funding statement: This work was supported in part by the National Science Foundation under Grant No. DMS-19-13035.

A Interior Regularity of ˉz

We will establish (2.4) by relating (2.3) to fourth-order variational inequalities analyzed in [26, 27, 18].

It follows from (2.3) and the Riesz representation theorem for non-negative functionals (cf. [43, 46, 25]) that

(A.1) ÎČ[(Lˉz,Lz)L2(Ω)+(g,Lz)L2(Ω)]+(ˉz-zd,z)L2(Ω)=∫ΩzdΜ for allz∈̊E(Δ;L2(Ω)),

where 𝜈 is a non-positive regular Borel measure. Moreover,

(A.2) Îœis supported onA={x∈Ω:ˉz=˜ψ(x)}={x∈Ω:ˉy(x)=ψ(x)}

by the principle of virtual work.

Let 𝜙 be any C∞ function with compact support in Ω such that

(A.3) ϕ=1 on an open neighborhood ofA.

We will show that ˜z=ϕˉz belongs to H3(Ω)∩W2∞(Ω) , which then implies (2.4).

Given any z∈̊E(Δ;L2(Ω)) , we have, in view of (1.7),

(A.4) L(ϕz)=-Δ(ϕz)+ζ⋅∇(ϕz)+Îł(ϕz)=-(Δϕ)z-2∇ϕ⋅∇z-ϕ(Δz)+(ζ⋅∇ϕ)z+ϕ(ζ⋅∇z)+ϕ(Îłz)=ϕLz+Mz,

where M:̊E(Δ;L2(Ω))→H10(Ω) is defined by

(A.5) Mz=-2∇ϕ⋅∇z+(ζ⋅∇ϕ-Δϕ)z.

Here Mz∈H10(Ω) because ̊E(Δ;L2(Ω)) is a subspace of H2loc(Ω) .

Note that ˜z belongs to

(A.6) K†={z†∈H20(Ω):z†≀ϕ˜ψonΩ},

and we have, in view of (A.2) and (A.3),

(A.7) ∫Ωϕ(z†-˜z)dÎœ=∫A(z†-ϕ˜ψ)dΜ≄0 for allz†∈K†.

It follows from (A.1), (A.4) and (A.7) that

(L˜z,L(z†-˜z))L2(Ω)=(Lˉz,ϕL(z†-˜z))L2(Ω)+(Mˉz,L(z†-˜z))L2(Ω)=(Lˉz,L(ϕ(z†-˜z)))L2(Ω)-(Lˉz,M(z†-˜z))L2(Ω)+(Mˉz,L(z†-˜z))L2(Ω)≄-ÎČ-1(ˉz-zd,ϕ(z†-˜z))L2(Ω)-(g,L(ϕ(z†-˜z)))L2(Ω)-(Lˉz,M(z†-˜z))L2(Ω)+(Mˉz,L(z†-˜z))L2(Ω),

which together with (1.7) implies

(A.8) (Δ˜z,Δ(z†-˜z))L2(Ω)≄-ÎČ-1(ˉz-zd,ϕ(z†-˜z))L2(Ω)-(Lˉz,M(z†-˜z))L2(Ω)-(g,-Δ(ϕ(z†-˜z))+(ζ⋅∇)(ϕ(z†-˜z))+ÎłÏ•(z†-˜z))L2(Ω)+(Mˉz,-Δ(z†-˜z)+ζ⋅∇(z†-˜z)+Îł(z†-˜z))L2(Ω)+(Δ˜z,ζ⋅∇(z†-˜z)+Îł(z†-˜z))L2(Ω)-(ζ⋅∇˜z+γ˜z,-Δ(z†-˜z)+ζ⋅∇(z†-˜z)+Îł(z†-˜z))L2(Ω).

Since ˜z=ϕˉz belongs to H20(Ω) , Mˉz belongs to H10(Ω) , 𝑔 belongs to H4(Ω) , đœ» belongs to [W1∞(Ω)]n and đ›Ÿ belongs to W1∞(Ω) , we can use (A.5) and integration by parts to rewrite (A.8) in the form of

(A.9) (Δ˜z,Δ(z†-˜z))L2(Ω)≄n∑i=1(fi,∂i(z†-˜z))L2(Ω)+(f0,z†-˜z)L2(Ω) for allz†∈K†,

where fi∈L2(Ω) for 0≀i≀n .

Note that (A.6) and (A.9) define a biharmonic variational inequality treated in [26]. Therefore, we can apply the interior regularity result there to conclude that ˜z∈H3loc(Ω) , and hence ˜z∈H3(Ω) because ˜z is compactly supported in Ω. We can also conclude that ˉz∈H3loc(Ω) .

According to the Sobolev embedding theorem, we have H1(Ω)â†ȘL6(Ω) and W16/5(Ω)â†ȘL2(Ω) in both two and three dimensions. Hence we can use (A.4), the facts that ζ∈[W1∞(Ω)]n , γ∈W1∞(Ω) , ˉz∈H3loc(Ω) together with integration by parts to rewrite (A.8) in the form of

(A.10) (Δ˜z,Δ(z†-˜z))L2(Ω)≄F(z†-˜z),

where F∈W-16(Ω) .

Let ρ∈H20(Ω) be defined by

(Δρ,Δv)L2(Ω)=F(v) for allv∈H20(Ω).

Then 𝜌 belongs to W36,loc(Ω)⊂W2∞,loc(Ω) by interior elliptic regularity (cf. [2, section 14]) and the Sobolev embedding theorem, and (A.10) becomes the variational inequality

(A.11) (Δz*,Δ(z♯-z*))L2(Ω)≄0 for allz♯∈K♯,

where K♯={z♯∈H20(Ω):zâ™Żâ‰€Ï•ËœÏˆ-ρ} and z*=˜z-ρ∈K♯ .

We can now apply the interior regularity results in [27, 18] to the biharmonic variational inequality (A.11) to conclude that z*∈W2∞,loc(Ω) , and hence ˜z=z*+ρ∈W2∞(Ω) because ˜z is compactly supported in Ω.

B Estimates for Rhˉy

It follows from the assumptions on đœ» and đ›Ÿ that we have

(B.1) a(y,z)≀C∄y∄H1(Ω)∄z∄H1(Ω) for ally,z∈H1(Ω),

and also the following GĂ„rding inequality (cf. [11, Theorem 5.6.8]):

a(z,z)+Îș∄z∄2L2(Ω)≄12∄z∄2H1(Ω) for allz∈H1(Ω),

where 𝜅 is a positive constant.

Recall Ih:C(ˉΩ)→Vh is the nodal interpolation operator and there is a standard estimate (cf. [20, 23, 11])

(B.2) |ζ-Ihζ|Hs(T)≀Cht-sT|ζ|Ht(T)

that holds for t>n2 , 0≀s≀t , ζ∈Ht(T) and T∈Th .

In view of (1.9) and (B.2), we have the following interpolation error estimate (cf. [20, 3, 23, 30, 11]):

(B.3) ∄z-Ihz∄L2(Ω)+h|z-Ihz|H1(Ω)≀Ch1+τ∄Δz∄L2(Ω) for allz∈̊E(Δ;L2(Ω)),

where 𝜏 is defined in (3.2). It follows from (B.3) that

(B.4) ∄ˉy-Ihˉy∄L2(Ω)+h|ˉy-Ihˉy|H1(Ω)≀Ch1+τ

because ˉy∈g+̊E(Δ;L2(Ω)) and g∈H4(Ω) . As mentioned in Remark 3.3, the finite element approximation Rhˉy is well-defined for ℎ sufficiently small.

Since the function Ihˉy-Rhˉy belongs to ̊Vh⊂H10(Ω) , we have

(B.5) 12∄Ihˉy-Rhˉy∄2H1(Ω)≀a(Ihˉy-Rhˉy,Ihˉy-Rhˉy)+Îș∄Ihˉy-Rhˉy∄2L2(Ω)=a(Ihˉy-ˉy,Ihˉy-Rhˉy)+Îș∄Ihˉy-Rhˉy∄2L2(Ω)≀Chτ∄Ihˉy-Rhˉy∄H1(Ω)+Îș∄Ihˉy-Rhˉy∄2L2(Ω)

by (3.3), (B.1) and (B.4).

Let ϕ∈H10(Ω) be defined by

(B.6) a(z,ϕ)=(z,Ihˉy-Rhˉy)L2(Ω) for allz∈H10(Ω).

Then 𝜙 belongs to ̊E(Δ;L2(Ω)) , and we have

(B.7) ∄ϕ∄H1+α(Ω)≀C∄Δϕ∄L2(Ω)≀C∄Ihˉy-Rhˉy∄L2(Ω)

by elliptic regularity.

It follows from (3.3) and (B.6) that

(B.8) ∄Ihˉy-Rhˉy∄2L2(Ω)=a(Ihˉy-Rhˉy,ϕ-Ihϕ)+a(Ihˉy-ˉy,Ihϕ),

and we can use (B.1), (B.3) and (B.7) to estimate the first term on the right-hand side of (B.8) by

(B.9) a(Ihˉy-Rhˉy,ϕ-Ihϕ)≀Chτ∄Ihˉy-Rhˉy∄H1(Ω)∄Ihˉy-Rhˉy∄L2(Ω).

According to (1.4), the second term on the right-hand side of (B.8) is given by

(B.10) a(Ihˉy-ˉy,Ihϕ)=∫Ω∇(Ihˉy-ˉy)⋅∇(Ihϕ)dx+∫Ω[ζ⋅∇(Ihˉy-ˉy)]Ihϕdx+∫Ωγ(Ihˉy-ˉy)Ihϕdx,

and we have

(B.11) ∫Ω[ζ⋅∇(Ihˉy-ˉy)]Ihϕdx+∫Ωγ(Ihˉy-ˉy)Ihϕdx=-∫Ω(Ihˉy-ˉy)ζ⋅∇(Ihϕ)dx+∫Ω(Îł-∇⋅ζ)(Ihˉy-ˉy)Ihϕdx≀Ch1+τ∄Ihˉy-Rhˉy∄L2(Ω)≀Ch2τ∄Ihˉy-Rhˉy∄L2(Ω)

by (B.3) and (B.7).

It only remains to estimate the first term on the right-hand side of (B.10), which can be rewritten through integration by parts as

(B.12) ∫Ω∇(Ihˉy-ˉy)⋅∇(Ihϕ)dx=∑σ∈Sh∫σ(Ihˉy-ˉy)⟩

where S h is the set of all the sides, ⟩ ∂ ⁥ ( I h ⁹ ϕ - ϕ ) / ∂ ⁥ n ⟧ is the jump of the normal derivative of ( I h ⁹ ϕ - ϕ ) across 𝜎, and d ⁹ S denotes the infinitesimal length ( n = 2 ) or infinitesimal area ( n = 3 ).

Lemma B.1

We have

(B.13) ∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x ≀ C ⁹ h 2 ⁹ τ ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) .

Proof

Let 𝜎 be a side (edge if n = 2 and face if n = 3 ) of the element 𝑇. By the trace theorem with scaling, we have, for 1 2 < s ≀ 1 ,

(B.14) ∄ ζ ∄ L 2 ⁹ ( σ ) ≀ C ⁹ [ h T - 1 / 2 ⁹ ∄ ζ ∄ L 2 ⁹ ( T ) + h T s - ( 1 / 2 ) ⁹ | ζ | H s ⁹ ( T ) ] .

In the case of quasi-uniform meshes, we can use (B.2), (B.7), (B.12) and (B.14) to obtain

∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x ≀ C ⁹ ∑ T ∈ T h h T α / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H ( 1 + α ) / 2 ⁹ ( T ) ⁹ h T α - ( 1 / 2 ) ⁹ | ϕ | H 1 + α ⁹ ( T )
≀ C ⁹ h 2 ⁹ α ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ,
which is (B.13) for quasi-uniform meshes.

The case of graded meshes in two dimensions is more involved. Let c 1 , 
 , c L be the corners of Ω, and let ω ℓ be the interior angle at c ℓ . We take α ℓ to be a number less than π ω ( α ℓ = 1 if ω < π ) so that the index of elliptic regularity α = min 1 ≀ ℓ ≀ L ⁥ α ℓ .

We can use (B.12) and (B.14) to obtain

(B.15) ∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x ≀ C ⁹ ∑ ℓ = 1 L ∑ T ∈ T h , ℓ h T α ℓ / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H ( 1 + α ℓ ) / 2 ⁹ ( T ) ⁹ h T α ℓ - ( 1 / 2 ) ⁹ | ϕ | H 1 + α ℓ ⁹ ( T ) + C ⁹ ∑ T ∈ T ~ h h T 1 / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H 1 ⁹ ( T ) ⁹ h T 1 / 2 ⁹ | ϕ | H 2 ⁹ ( T ) ,

where T h , ℓ is the set of the triangles in T h that touch the corner c ℓ , and T ~ h = T h ∖ ( ⋃ ℓ = 1 L T h , ℓ ) . Note that y ¯ and 𝜙 belong to H 2 ⁱ ( T ) for T ∈ T ~ h (cf. Remark 1.4).

The first sum on the right-hand side of (B.15) is bounded by

∑ ℓ = 1 L ∑ T ∈ T h , ℓ h T α ℓ / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H ( 1 + α ℓ ) / 2 ⁹ ( T ) ⁹ h T α ℓ - ( 1 / 2 ) ⁹ | ϕ | H 1 + α ℓ ⁹ ( T ) ≀ C ⁹ ∑ ℓ = 1 L ∑ T ∈ T h , ℓ h T 2 ⁹ α ℓ ⁹ | y ÂŻ | H 1 + α ℓ ⁹ ( T ) ⁹ | ϕ | H 1 + α ℓ ⁹ ( T ) ≀ C ⁹ h 2 ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ,

where we have used (B.2), (B.7) and the fact that, on the graded mesh, we have h T ≈ h 1 / α ℓ if T ∈ T h , ℓ (cf. [3, Section 4] and [30, Section 8.4.1]).

Finally, the second sum on the right-hand side of (B.15) is bounded by

∑ T ∈ T ~ h h T 1 / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H 1 ⁹ ( T ) ⁹ h T 1 / 2 ⁹ | ϕ | H 2 ⁹ ( T ) ≀ C ⁹ ∑ T ∈ T ~ h h T 2 ⁹ | y ÂŻ | H 2 ⁹ ( T ) ⁹ | ϕ | H 2 ⁹ ( T ) ≀ C ⁹ h 2 ⁹ ( ∑ T ∈ T ~ h ( h T / h ) 2 ⁹ | y ÂŻ | H 2 ⁹ ( T ) 2 ) 1 / 2 ⁹ ( ∑ T ∈ T ~ h ( h T / h ) 2 ⁹ | ϕ | H 2 ⁹ ( T ) ) 1 / 2 ≀ C ⁹ h 2 ⁹ ∄ Δ ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ⁹ ∄ Δ ⁹ ϕ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h 2 ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ,

where we have used (B.2), (B.7), the fact that, on the graded mesh, we have

( h T / h ) ≈ ( distance between ⁹ T ⁹ and the closest corner ⁹ c ℓ ) 1 - α ℓ   if ⁹ T ∈ T ~ h ,

together with the nature of the singularity at a reentrant corner of Ω (cf. [3, Section 4] and [30, Section 8.4.1]). ∎

Putting (B.8)–(B.11) and (B.13) together, we arrive at the estimate

(B.16) ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h τ ⁹ ( h τ + ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) ) .

It follows from (B.5) and (B.16) that

∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) ≀ C ⁹ h τ   and   ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h 2 ⁹ τ ,

which together with (B.3) imply (3.4) and (3.5).

Finally, estimate (3.6) follows from (2.6), (3.5) and the interior maximum norm estimate in [45, equation (0.8)].

Acknowledgements

The authors would like to thank Joscha Gedicke for helpful discussions concerning the numerical examples.

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Received: 2021-05-22
Accepted: 2021-05-31
Published Online: 2021-06-23
Published in Print: 2021-10-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Abstract

We investigate a P 1 finite element method for an elliptic distributed optimal control problem with pointwise state constraints and a state equation that includes advective/convective and reactive terms. The convergence of this method can be established for general polygonal/polyhedral domains that are not necessarily convex. The discrete problem is a strictly convex quadratic program with box constraints that can be solved efficiently by a primal-dual active set algorithm.

Keywords: Elliptic Distributed Optimal Control Problems; General State Equation; Pointwise State Constraints
MSC 2010: 65N30; 65K15; 90C20

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-19-13035

Funding statement: This work was supported in part by the National Science Foundation under Grant No. DMS-19-13035.

A Interior Regularity of z ÂŻ

We will establish (2.4) by relating (2.3) to fourth-order variational inequalities analyzed in [26, 27, 18].

It follows from (2.3) and the Riesz representation theorem for non-negative functionals (cf. [43, 46, 25]) that

(A.1) ÎČ âą [ ( L ⁹ z ÂŻ , L ⁹ z ) L 2 ⁹ ( Ω ) + ( g , L ⁹ z ) L 2 ⁹ ( Ω ) ] + ( z ÂŻ - z d , z ) L 2 ⁹ ( Ω ) = ∫ Ω z ⁹ d Îœ   for all ⁹ z ∈ E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) ,

where 𝜈 is a non-positive regular Borel measure. Moreover,

(A.2) Îœ ⁹ is supported on ⁹ A = { x ∈ Ω : z ÂŻ = ψ ~ ⁹ ( x ) } = { x ∈ Ω : y ÂŻ ⁹ ( x ) = ψ ⁹ ( x ) }

by the principle of virtual work.

Let 𝜙 be any C ∞ function with compact support in Ω such that

(A.3) ϕ = 1   on an open neighborhood of ⁹ A .

We will show that z ~ = ϕ ⁹ z ÂŻ belongs to H 3 ⁹ ( Ω ) ∩ W ∞ 2 ⁹ ( Ω ) , which then implies (2.4).

Given any z ∈ E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) , we have, in view of (1.7),

(A.4) L ⁹ ( ϕ ⁹ z ) = - Δ ⁹ ( ϕ ⁹ z ) + ζ ⋅ ∇ ⁥ ( ϕ ⁹ z ) + Îł ⁹ ( ϕ ⁹ z ) = - ( Δ ⁹ ϕ ) ⁹ z - 2 ⁹ ∇ ⁥ ϕ ⋅ ∇ ⁥ z - ϕ ⁹ ( Δ ⁹ z ) + ( ζ ⋅ ∇ ⁥ ϕ ) ⁹ z + ϕ ⁹ ( ζ ⋅ ∇ ⁥ z ) + ϕ ⁹ ( Îł ⁹ z ) = ϕ ⁹ L ⁹ z + M ⁹ z ,

where M : E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) → H 0 1 ⁹ ( Ω ) is defined by

(A.5) M ⁹ z = - 2 ⁹ ∇ ⁥ ϕ ⋅ ∇ ⁥ z + ( ζ ⋅ ∇ ⁥ ϕ - Δ ⁹ ϕ ) ⁹ z .

Here M ⁹ z ∈ H 0 1 ⁹ ( Ω ) because E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) is a subspace of H loc 2 ⁹ ( Ω ) .

Note that z ~ belongs to

(A.6) K † = { z † ∈ H 0 2 ⁹ ( Ω ) : z † ≀ ϕ ⁹ ψ ~ ⁹ on ⁹ Ω } ,

and we have, in view of (A.2) and (A.3),

(A.7) ∫ Ω ϕ ⁹ ( z † - z ~ ) ⁹ d Îœ = ∫ A ( z † - ϕ ⁹ ψ ~ ) ⁹ d Îœ ≄ 0   for all ⁹ z † ∈ K † .

It follows from (A.1), (A.4) and (A.7) that

( L ⁹ z ~ , L ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) = ( L ⁹ z ÂŻ , ϕ ⁹ L ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) + ( M ⁹ z ÂŻ , L ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) = ( L ⁹ z ÂŻ , L ⁹ ( ϕ ⁹ ( z † - z ~ ) ) ) L 2 ⁹ ( Ω ) - ( L ⁹ z ÂŻ , M ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) + ( M ⁹ z ÂŻ , L ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) ≄ - ÎČ - 1 ⁹ ( z ÂŻ - z d , ϕ ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) - ( g , L ⁹ ( ϕ ⁹ ( z † - z ~ ) ) ) L 2 ⁹ ( Ω ) - ( L ⁹ z ÂŻ , M ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) + ( M ⁹ z ÂŻ , L ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) ,

which together with (1.7) implies

(A.8) ( Δ ⁹ z ~ , Δ ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) ≄ - ÎČ - 1 ⁹ ( z ÂŻ - z d , ϕ ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) - ( L ⁹ z ÂŻ , M ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) - ( g , - Δ ⁹ ( ϕ ⁹ ( z † - z ~ ) ) + ( ζ ⋅ ∇ ) ⁹ ( ϕ ⁹ ( z † - z ~ ) ) + Îł ⁹ ϕ ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) + ( M ⁹ z ÂŻ , - Δ ⁹ ( z † - z ~ ) + ζ ⋅ ∇ ⁥ ( z † - z ~ ) + Îł ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) + ( Δ ⁹ z ~ , ζ ⋅ ∇ ⁥ ( z † - z ~ ) + Îł ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) - ( ζ ⋅ ∇ ⁥ z ~ + Îł ⁹ z ~ , - Δ ⁹ ( z † - z ~ ) + ζ ⋅ ∇ ⁥ ( z † - z ~ ) + Îł ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) .

Since z ~ = ϕ ⁹ z ÂŻ belongs to H 0 2 ⁹ ( Ω ) , M ⁹ z ÂŻ belongs to H 0 1 ⁹ ( Ω ) , 𝑔 belongs to H 4 ⁹ ( Ω ) , đœ» belongs to [ W ∞ 1 ⁹ ( Ω ) ] n and đ›Ÿ belongs to W ∞ 1 ⁹ ( Ω ) , we can use (A.5) and integration by parts to rewrite (A.8) in the form of

(A.9) ( Δ ⁹ z ~ , Δ ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) ≄ ∑ i = 1 n ( f i , ∂ i ⁥ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) + ( f 0 , z † - z ~ ) L 2 ⁹ ( Ω )   for all ⁹ z † ∈ K † ,

where f i ∈ L 2 ⁹ ( Ω ) for 0 ≀ i ≀ n .

Note that (A.6) and (A.9) define a biharmonic variational inequality treated in [26]. Therefore, we can apply the interior regularity result there to conclude that z ~ ∈ H loc 3 ⁹ ( Ω ) , and hence z ~ ∈ H 3 ⁹ ( Ω ) because z ~ is compactly supported in Ω. We can also conclude that z ¯ ∈ H loc 3 ⁹ ( Ω ) .

According to the Sobolev embedding theorem, we have H 1 ⁹ ( Ω ) â†Ș L 6 ⁹ ( Ω ) and W 6 / 5 1 ⁹ ( Ω ) â†Ș L 2 ⁹ ( Ω ) in both two and three dimensions. Hence we can use (A.4), the facts that ζ ∈ [ W ∞ 1 ⁹ ( Ω ) ] n , Îł ∈ W ∞ 1 ⁹ ( Ω ) , z ÂŻ ∈ H loc 3 ⁹ ( Ω ) together with integration by parts to rewrite (A.8) in the form of

(A.10) ( Δ ⁹ z ~ , Δ ⁹ ( z † - z ~ ) ) L 2 ⁹ ( Ω ) ≄ F ⁹ ( z † - z ~ ) ,

where F ∈ W 6 - 1 ⁹ ( Ω ) .

Let ρ ∈ H 0 2 ⁹ ( Ω ) be defined by

( Δ ⁹ ρ , Δ ⁹ v ) L 2 ⁹ ( Ω ) = F ⁹ ( v )   for all ⁹ v ∈ H 0 2 ⁹ ( Ω ) .

Then 𝜌 belongs to W 6 , l ⁹ o ⁹ c 3 ⁹ ( Ω ) ⊂ W ∞ , loc 2 ⁹ ( Ω ) by interior elliptic regularity (cf. [2, section 14]) and the Sobolev embedding theorem, and (A.10) becomes the variational inequality

(A.11) ( Δ ⁹ z * , Δ ⁹ ( z ♯ - z * ) ) L 2 ⁹ ( Ω ) ≄ 0   for all ⁹ z ♯ ∈ K ♯ ,

where K ♯ = { z ♯ ∈ H 0 2 ⁹ ( Ω ) : z ♯ ≀ ϕ ⁹ ψ ~ - ρ } and z * = z ~ - ρ ∈ K ♯ .

We can now apply the interior regularity results in [27, 18] to the biharmonic variational inequality (A.11) to conclude that z * ∈ W ∞ , loc 2 ⁹ ( Ω ) , and hence z ~ = z * + ρ ∈ W ∞ 2 ⁹ ( Ω ) because z ~ is compactly supported in Ω.

B Estimates for R h ⁹ y ¯

It follows from the assumptions on đœ» and đ›Ÿ that we have

(B.1) a ⁹ ( y , z ) ≀ C ⁹ ∄ y ∄ H 1 ⁹ ( Ω ) ⁹ ∄ z ∄ H 1 ⁹ ( Ω )   for all ⁹ y , z ∈ H 1 ⁹ ( Ω ) ,

and also the following GĂ„rding inequality (cf. [11, Theorem 5.6.8]):

a ⁹ ( z , z ) + Îș ⁹ ∄ z ∄ L 2 ⁹ ( Ω ) 2 ≄ 1 2 ⁹ ∄ z ∄ H 1 ⁹ ( Ω ) 2   for all ⁹ z ∈ H 1 ⁹ ( Ω ) ,

where 𝜅 is a positive constant.

Recall I h : C ⁹ ( Ω ÂŻ ) → V h is the nodal interpolation operator and there is a standard estimate (cf. [20, 23, 11])

(B.2) | ζ - I h ⁹ ζ | H s ⁹ ( T ) ≀ C ⁹ h T t - s ⁹ | ζ | H t ⁹ ( T )

that holds for t > n 2 , 0 ≀ s ≀ t , ζ ∈ H t ⁹ ( T ) and T ∈ T h .

In view of (1.9) and (B.2), we have the following interpolation error estimate (cf. [20, 3, 23, 30, 11]):

(B.3) ∄ z - I h ⁹ z ∄ L 2 ⁹ ( Ω ) + h ⁹ | z - I h ⁹ z | H 1 ⁹ ( Ω ) ≀ C ⁹ h 1 + τ ⁹ ∄ Δ ⁹ z ∄ L 2 ⁹ ( Ω )   for all ⁹ z ∈ E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) ,

where 𝜏 is defined in (3.2). It follows from (B.3) that

(B.4) ∄ y ÂŻ - I h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) + h ⁹ | y ÂŻ - I h ⁹ y ÂŻ | H 1 ⁹ ( Ω ) ≀ C ⁹ h 1 + τ

because y ÂŻ ∈ g + E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) and g ∈ H 4 ⁹ ( Ω ) . As mentioned in Remark 3.3, the finite element approximation R h ⁹ y ÂŻ is well-defined for ℎ sufficiently small.

Since the function I h ⁹ y ÂŻ - R h ⁹ y ÂŻ belongs to V ̊ h ⊂ H 0 1 ⁹ ( Ω ) , we have

(B.5) 1 2 ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) 2 ≀ a ⁹ ( I h ⁹ y ÂŻ - R h ⁹ y ÂŻ , I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ) + Îș ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) 2 = a ⁹ ( I h ⁹ y ÂŻ - y ÂŻ , I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ) + Îș ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) 2 ≀ C ⁹ h τ ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) + Îș ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) 2

by (3.3), (B.1) and (B.4).

Let ϕ ∈ H 0 1 ⁹ ( Ω ) be defined by

(B.6) a ⁹ ( z , ϕ ) = ( z , I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ) L 2 ⁹ ( Ω )   for all ⁹ z ∈ H 0 1 ⁹ ( Ω ) .

Then 𝜙 belongs to E ̊ ⁹ ( Δ ; L 2 ⁹ ( Ω ) ) , and we have

(B.7) ∄ ϕ ∄ H 1 + α ⁹ ( Ω ) ≀ C ⁹ ∄ Δ ⁹ ϕ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω )

by elliptic regularity.

It follows from (3.3) and (B.6) that

(B.8) ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) 2 = a ⁹ ( I h ⁹ y ÂŻ - R h ⁹ y ÂŻ , ϕ - I h ⁹ ϕ ) + a ⁹ ( I h ⁹ y ÂŻ - y ÂŻ , I h ⁹ ϕ ) ,

and we can use (B.1), (B.3) and (B.7) to estimate the first term on the right-hand side of (B.8) by

(B.9) a ⁹ ( I h ⁹ y ÂŻ - R h ⁹ y ÂŻ , ϕ - I h ⁹ ϕ ) ≀ C ⁹ h τ ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) .

According to (1.4), the second term on the right-hand side of (B.8) is given by

(B.10) a ⁹ ( I h ⁹ y ÂŻ - y ÂŻ , I h ⁹ ϕ ) = ∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x + ∫ Ω [ ζ ⋅ ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ] ⁹ I h ⁹ ϕ ⁹ d x + ∫ Ω Îł ⁹ ( I h ⁹ y ÂŻ - y ÂŻ ) ⁹ I h ⁹ ϕ ⁹ d x ,

and we have

(B.11) ∫ Ω [ ζ ⋅ ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ] ⁹ I h ⁹ ϕ ⁹ d x + ∫ Ω Îł ⁹ ( I h ⁹ y ÂŻ - y ÂŻ ) ⁹ I h ⁹ ϕ ⁹ d x = - ∫ Ω ( I h ⁹ y ÂŻ - y ÂŻ ) ⁹ ζ ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x + ∫ Ω ( Îł - ∇ ⋅ ζ ) ⁹ ( I h ⁹ y ÂŻ - y ÂŻ ) ⁹ I h ⁹ ϕ ⁹ d x ≀ C ⁹ h 1 + τ ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h 2 ⁹ τ ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω )

by (B.3) and (B.7).

It only remains to estimate the first term on the right-hand side of (B.10), which can be rewritten through integration by parts as

(B.12) ∫ Ω ∇ ( I h y ÂŻ - y ÂŻ ) ⋅ ∇ ( I h ϕ ) d x = ∑ σ ∈ S h ∫ σ ( I h y ÂŻ - y ÂŻ ) ⟩ ∂ ( I h ϕ - ϕ ) / ∂ n ⟧ d S ,

where S h is the set of all the sides, ⟩ ∂ ⁥ ( I h ⁹ ϕ - ϕ ) / ∂ ⁥ n ⟧ is the jump of the normal derivative of ( I h ⁹ ϕ - ϕ ) across 𝜎, and d ⁹ S denotes the infinitesimal length ( n = 2 ) or infinitesimal area ( n = 3 ).

Lemma B.1

We have

(B.13) ∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x ≀ C ⁹ h 2 ⁹ τ ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) .

Proof

Let 𝜎 be a side (edge if n = 2 and face if n = 3 ) of the element 𝑇. By the trace theorem with scaling, we have, for 1 2 < s ≀ 1 ,

(B.14) ∄ ζ ∄ L 2 ⁹ ( σ ) ≀ C ⁹ [ h T - 1 / 2 ⁹ ∄ ζ ∄ L 2 ⁹ ( T ) + h T s - ( 1 / 2 ) ⁹ | ζ | H s ⁹ ( T ) ] .

In the case of quasi-uniform meshes, we can use (B.2), (B.7), (B.12) and (B.14) to obtain

∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x ≀ C ⁹ ∑ T ∈ T h h T α / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H ( 1 + α ) / 2 ⁹ ( T ) ⁹ h T α - ( 1 / 2 ) ⁹ | ϕ | H 1 + α ⁹ ( T )
≀ C ⁹ h 2 ⁹ α ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ,
which is (B.13) for quasi-uniform meshes.

The case of graded meshes in two dimensions is more involved. Let c 1 , 
 , c L be the corners of Ω, and let ω ℓ be the interior angle at c ℓ . We take α ℓ to be a number less than π ω ( α ℓ = 1 if ω < π ) so that the index of elliptic regularity α = min 1 ≀ ℓ ≀ L ⁥ α ℓ .

We can use (B.12) and (B.14) to obtain

(B.15) ∫ Ω ∇ ⁥ ( I h ⁹ y ÂŻ - y ÂŻ ) ⋅ ∇ ⁥ ( I h ⁹ ϕ ) ⁥ d ⁹ x ≀ C ⁹ ∑ ℓ = 1 L ∑ T ∈ T h , ℓ h T α ℓ / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H ( 1 + α ℓ ) / 2 ⁹ ( T ) ⁹ h T α ℓ - ( 1 / 2 ) ⁹ | ϕ | H 1 + α ℓ ⁹ ( T ) + C ⁹ ∑ T ∈ T ~ h h T 1 / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H 1 ⁹ ( T ) ⁹ h T 1 / 2 ⁹ | ϕ | H 2 ⁹ ( T ) ,

where T h , ℓ is the set of the triangles in T h that touch the corner c ℓ , and T ~ h = T h ∖ ( ⋃ ℓ = 1 L T h , ℓ ) . Note that y ¯ and 𝜙 belong to H 2 ⁱ ( T ) for T ∈ T ~ h (cf. Remark 1.4).

The first sum on the right-hand side of (B.15) is bounded by

∑ ℓ = 1 L ∑ T ∈ T h , ℓ h T α ℓ / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H ( 1 + α ℓ ) / 2 ⁹ ( T ) ⁹ h T α ℓ - ( 1 / 2 ) ⁹ | ϕ | H 1 + α ℓ ⁹ ( T ) ≀ C ⁹ ∑ ℓ = 1 L ∑ T ∈ T h , ℓ h T 2 ⁹ α ℓ ⁹ | y ÂŻ | H 1 + α ℓ ⁹ ( T ) ⁹ | ϕ | H 1 + α ℓ ⁹ ( T ) ≀ C ⁹ h 2 ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ,

where we have used (B.2), (B.7) and the fact that, on the graded mesh, we have h T ≈ h 1 / α ℓ if T ∈ T h , ℓ (cf. [3, Section 4] and [30, Section 8.4.1]).

Finally, the second sum on the right-hand side of (B.15) is bounded by

∑ T ∈ T ~ h h T 1 / 2 ⁹ | I h ⁹ y ÂŻ - y ÂŻ | H 1 ⁹ ( T ) ⁹ h T 1 / 2 ⁹ | ϕ | H 2 ⁹ ( T ) ≀ C ⁹ ∑ T ∈ T ~ h h T 2 ⁹ | y ÂŻ | H 2 ⁹ ( T ) ⁹ | ϕ | H 2 ⁹ ( T ) ≀ C ⁹ h 2 ⁹ ( ∑ T ∈ T ~ h ( h T / h ) 2 ⁹ | y ÂŻ | H 2 ⁹ ( T ) 2 ) 1 / 2 ⁹ ( ∑ T ∈ T ~ h ( h T / h ) 2 ⁹ | ϕ | H 2 ⁹ ( T ) ) 1 / 2 ≀ C ⁹ h 2 ⁹ ∄ Δ ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ⁹ ∄ Δ ⁹ ϕ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h 2 ⁹ ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ,

where we have used (B.2), (B.7), the fact that, on the graded mesh, we have

( h T / h ) ≈ ( distance between ⁹ T ⁹ and the closest corner ⁹ c ℓ ) 1 - α ℓ   if ⁹ T ∈ T ~ h ,

together with the nature of the singularity at a reentrant corner of Ω (cf. [3, Section 4] and [30, Section 8.4.1]). ∎

Putting (B.8)–(B.11) and (B.13) together, we arrive at the estimate

(B.16) ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h τ ⁹ ( h τ + ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) ) .

It follows from (B.5) and (B.16) that

∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ H 1 ⁹ ( Ω ) ≀ C ⁹ h τ   and   ∄ I h ⁹ y ÂŻ - R h ⁹ y ÂŻ ∄ L 2 ⁹ ( Ω ) ≀ C ⁹ h 2 ⁹ τ ,

which together with (B.3) imply (3.4) and (3.5).

Finally, estimate (3.6) follows from (2.6), (3.5) and the interior maximum norm estimate in [45, equation (0.8)].

Acknowledgements

The authors would like to thank Joscha Gedicke for helpful discussions concerning the numerical examples.

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Received: 2021-05-22
Accepted: 2021-05-31
Published Online: 2021-06-23
Published in Print: 2021-10-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors
  3. A 𝑃1 Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints
  4. A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain
  5. Analysis of Backward Euler Primal DPG Methods
  6. A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations
  7. Error Estimation and Adaptivity for Differential Equations with Multiple Scales in Time
  8. A Shift Splitting Iteration Method for Generalized Absolute Value Equations
  9. Reconstruction of a Space-Dependent Coefficient in a Linear Benjamin–Bona–Mahony Type Equation
  10. Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem
  11. Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions
  12. Novel Adaptive Hybrid Discontinuous Galerkin Algorithms for Elliptic Problems

Articles in the same Issue

  1. Frontmatter
  2. A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors
  3. A 𝑃1 Finite Element Method for a Distributed Elliptic Optimal Control Problem with a General State Equation and Pointwise State Constraints
  4. A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain
  5. Analysis of Backward Euler Primal DPG Methods
  6. A Low-Dimensional Compact Finite Difference Method on Graded Meshes for Time-Fractional Diffusion Equations
  7. Error Estimation and Adaptivity for Differential Equations with Multiple Scales in Time
  8. A Shift Splitting Iteration Method for Generalized Absolute Value Equations
  9. Reconstruction of a Space-Dependent Coefficient in a Linear Benjamin–Bona–Mahony Type Equation
  10. Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem
  11. Two Finite Difference Schemes for Multi-Dimensional Fractional Wave Equations with Weakly Singular Solutions
  12. Novel Adaptive Hybrid Discontinuous Galerkin Algorithms for Elliptic Problems
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