# Superconvergence of Semidiscrete Splitting Positive Definite Mixed Finite Elements for Hyperbolic Optimal Control Problems

Authors: Yuchun Hua; Yuelong Tang
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we consider semidiscrete splitting positive definite mixed finite element methods for optimal control problems governed by hyperbolic equations with integral constraints. The state and costate are approximated by the lowest order Raviart-Thomas mixed rectangular finite element, and the control is approximated by piecewise constant functions. We derive some convergence and superconvergence results for the control, the state and the adjoint state. A numerical example is provided to demonstrate our theoretical results.


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## Body

## 1. Introduction

There have been extensive studies in error estimates of standard finite element methods (FEMs) for optimal control problems (OCPs). The convergence or superconvergence results of standard FEMs for elliptic and parabolic OCPs can be found in [1–13], respectively.Because large temperature gradients during cooling or heating may lead to its destruction in temperature control problems, the gradient stands for Darcy velocity in flow control problems, stiffness optimization in nonlinear pantographic structures [14], and topology optimization of a cycloidal metamaterial [15]; their objective functionals contain not only the primal state variable but also its gradient. At this time, mixed finite element methods (MFEMs) will be a very good choice for solving this kind of OCPs. The convergence or superconvergence results of MFEMs for elliptic and parabolic OCPs can be found in [16–20], respectively. However, mixed finite element spaces have to satisfy the Ladyženskaja-Babuška-Brezzi (LBB) condition, which brings very little available approximation spaces and expensive computing costs.In order to avoid the limitation of LBB condition, a splitting positive definite MFEM was first proposed for solving miscible displacement of compressible flow in a porous medium [21]. Compared with the classic MFEMs, the main advantages of this method are that the original problems can be split into two independent symmetric positive definite subschemes and that the LBB condition is not necessary. Recently, splitting positive definite MFEMs have been used to solve hyperbolic equations [22, 23], elliptic OCPs [24], and parabolic OCPs [25]. To our best knowledge, most of the published papers on different FEMs or MFEMs for OCPs are focused on elliptic or parabolic cases. Although Xu in [26] established a priori error estimates and superconvergence results of splitting positive definite MFEM for pseudohyperbolic integrodifferential OCPs and Lu et al. in [27] derived the convergence of finite volume element method for nonlinear hyperbolic OCPs, there are very little studies on hyperbolic OCPs, .The goal of this paper is to investigate splitting positive definite MFEMs for hyperbolic OCPs and derive the convergence and superconvergence.We are interested in the following hyperbolic OCPs:(1)minu∈K12∫0Tp−pd2+y−yd2+u2dt,(2)cxyttx,t+divpx,t=fx,t+ux,t,x∈Ω,t∈J,(3)px,t=−Ax∇yx,t,x∈Ω,t∈J,(4)yx,t=0,x∈∂Ω,t∈J,(5)yx,0=y0x,ytx,0=y1x,x∈Ω,where Ω⊂R2 is a rectangle domain, J=0,T, pd∈L2J;L2Ω2, yd,f∈L2J;L2Ω, c∈W1,∞Ω with 0<cmin≤cx≤cmax and y0,y1∈H2Ω. The coefficient matrix Ax=aijx2×2∈W1,∞Ω¯;R2×2 is a symmetric matrix, and there are constants c1,c2>0 satisfying any vector X∈R2, c1XR22≤XtAX≤c2XR22. K is a set defined by
(6)K=u∈L2J;L2Ω:∫0T∫Ωudxdt≥0.In this paper, we adopt the standard notationWm,pΩ for Sobolev spaces on Ω with a norm ·m,p given by vm,pp=∑α≤mDαvLpΩp, a seminorm ·m,p given by vm,pp=∑α=mDαvLpΩp. For p=2, we set HmΩ=Wm,2Ω,H01Ω=v∈H1Ω:v∂Ω=0, and ·m=·m,2,·=·0,2. We denote by LsJ;Wm,pΩ the Banach space of all Ls integrable functions from J into Wm,pΩ with norm vLsJ;Wm,pΩ=∫0TvWm,pΩsdt1/sfors∈1,∞, and the standard modification for s=∞. For simplicity of presentation, we denote vLsJ;Wm,pΩ by vLsWm,p. Similarly, one can define the spaces H1J;Wm,pΩ and CkJ;Wm,pΩ. In addition, C denotes a general positive constant independent of h, where h is the spatial mesh-size for the control and state discretization.The plan of this paper is as follows. In Section2, we give an equivalent optimality conditions for the OCP (1)–(5) and construct its splitting positive definite mixed finite element approximation scheme. In Section 3, we derive the convergence for the control variable, the state variables, and the adjoint state variables. In Section 4, we derive the superconvergence properties between the RT projections and the approximation solutions of the control and the state variables. In the last section, we present a numerical example to illustrate our theoretical results.


## 2. Splitting Positive Definite MFEMs for OCPs

In this section, we shall construct a splitting positive definite mixed finite element approximation of the control problems (1)–(5). To fix the idea, we shall take the state spaces L=H2J;V and Q=H2J;W, where V and W are defined as follows.Let(7)V=Hdiv;Ω=v∈L2Ω2,divv∈L2Ω,W=L2Ω,and the inner products
(8)f1,f2=∫Ωf1f2,∀f1,f2∈L2Ω,φ,ψ=∑i=12φi,ψi,∀φ,ψ∈L2Ω2.Letb=1/cx, a mixed weak form of (2), and (3) can be given by
(9)A−1p,v=y,divv,∀v∈V,t∈J,(10)ytt,w+bdivp,w=bf,w+bu,w,∀w∈W,t∈J.As in [24], taking w=divv,∀v∈V in (10), (9) differentiating twice with respect to t, and then substituting the two resulting equations, we derive
(11)A−1ptt,v+bdivp,divv=bf,divv+bu,divv,∀v∈V,t∈J.By using (10) and (11), we get the following new mixed variational form:
(12)A−1ptt,v+bdivp,divv=bf,divv+bu,divv,∀v∈V,t∈J,(13)ytt,w+bdivp,w=bf,w+bu,w,∀w∈W,t∈J.It is easily seen that (12) is separated from (13) so that p can be solved independently from (12).We recast (1)–(5) as the following weak form: find p,y,u∈L×Q×K such that
(14)minu∈K12∫0Tp−pd2+y−yd2+u2dt,(15)A−1ptt,v+bdivp,divv=bf,divv+bu,divv,∀v∈V,t∈J,(16)px,0=−A∇y0x,ptx,0=−A∇y1x,∀x∈Ω,(17)ytt,w+bdivp,w=bf,w+bu,w,∀w∈W,t∈J,(18)yx,0=y0x,ytx,0=y1x,∀x∈Ω.It then follows from [28] that the optimal control problems (14)–(18) have a unique solution p,y,u, and that a triplet p,y,u is the solution of (14)–(18) if and only if there is a costate q,z∈L×Q such that p,y,q,z,u satisfies the following optimality conditions:
(19)A−1ptt,v+bdivp,divv=bf,divv+bu,divv,∀v∈V,t∈J,(20)px,0=−A∇y0x,ptx,0=−A∇y1x,∀x∈Ω,(21)ytt,w+bdivp,w=bf,w+bu,w,∀w∈W,t∈J,(22)yx,0=y0x,ytx,0=y1x,∀x∈Ω,(23)A−1qtt,v+bdivp,divv+bz,divv=−p−pd,v,∀v∈V,t∈J,(24)qx,T=0,qtx,T=0,∀x∈Ω,(25)ztt,w=−y−yd,w,∀w∈W,t∈J,(26)zx,T=0,ztx,T=0,∀x∈Ω,(27)∫0Tu−bz−bdivq,u~−udt≥0,∀u~∈K.The inequality (27) can be expressed as
(28)u=max0,−G¯+G,where G=bz+bdivq and G¯=∫0T∫ΩGdxdt/∣Ω∣×T.LetTh be a uniform rectangulation of the domain Ω, and he denotes the diameter of element e and h=maxe∈Thhe. Let Vh×Wh⊂V×W denote the lowest order Raviart-Thomas mixed finite element space [29, 30], namely
(29)Vh≔vh∈V:∀e∈Th,vhe∈Q1,0e×Q0,1e,Wh≔wh∈W:∀e∈Th,whe∈Q0,0e,where Qm,ne indicates the space of polynomials of degree no more than m and n in x1 and x2 on e, respectively. And the approximated space of control is given by
(30)Kh≔L2J;Wh∩K.We introduce two projection operators. First, we define the standardL2Ω-projection [29] Ph:W⟶Wh, which satisfies the following: for any ϕ∈W(31)Phϕ−ϕ,wh=0,∀wh∈Wh,(32)ϕ−Phϕ−s,ρ≤Ch1+sϕ1,ρ,s=0,1,2≤ρ≤∞,∀ϕ∈W1,ρΩ.Second, we recall the Fortin projection (see [29, 31]) Πh:V⟶Vh, which satisfies the following: for any q∈V(33)divΠhq−q,wh=0,∀wh∈Wh,(34)q−Πhq0,ρ≤Chq1,ρ,2≤ρ≤∞,∀q∈W1,ρΩ2,(35)divq−Πhq≤Chdivq1,∀divq∈H1Ω.Then the splitting positive definite mixed finite element discretization of (14)–(18) is as follows: find ph,yh,uh∈H2J;Vh×H2J;Wh×Kh such that
(36)minuh∈Kh12∫0Tph−pd2+yh−yd2+uh2dt,(37)A−1ph,tt,vh+bdivph,divvh=bf,divvh+buh,divvh,∀vh∈Vh,t∈J,(38)phx,0=Πhpx,0,ph,tx,0=Πhptx,0,∀x∈Ω,(39)yh,tt,wh+bdivph,wh=bf,wh+buh,wh,∀wh∈Wh,t∈J,(40)yhx,0=Phy0x,yh,tx,0=Phy1x,∀x∈Ω.The optimal control problems (36)–(40) again have a unique solution ph,yh,uh, and a triplet ph,yh,uh is the solution of (36)–(40) if and only if there is a costate qh,zh such that ph,yh,qh,zh,uh satisfies the following optimality conditions:
(41)A−1ph,tt,vh+bdivph,divvh=bf,divvh+buh,divvh,∀vh∈Vh,t∈J,(42)phx,0=Πhpx,0,ph,tx,0=Πhptx,0,∀x∈Ω,(43)yh,tt,wh+bdivph,wh=bf,wh+buh,wh,∀wh∈Wh,t∈J,(44)yhx,0=Phy0x,yh,tx,0=Phy0x,∀x∈Ω,(45)A−1qh,tt,vh+bdivqh,divvh+bzh,divvh=−ph−pd,vh,∀vh∈Vh,t∈J,(46)qhx,T=0,qh,tx,T=0,∀x∈Ω,(47)zh,tt,wh=−yh−yd,wh,∀wh∈Wh,t∈J,(48)zhx,T=0,zh,tx,T=0,∀x∈Ω,(49)∫0Tuh−bzh−bdivqh,u~h−uhdt≥0,∀u~h∈Kh.Similar to (28), we have
(50)uh=max0,−G¯h+Gh,where Gh=Phbzh+divqh and G¯h=∫0T∫ΩGhdxdt/∣Ω∣×T.


## 3. Convergence Analysis

In this section, we will derive the convergence of splitting positive definite MFEMs for hyperbolic OCPs. For∀u~∈K, we define the discrete state solution phu~,yhu~,qhu~,zhu~ associated with u~ which satisfies
(51)A−1ph,ttu~,vh+bdivphu~,divvh=bf,divvh+bu~,divvh,∀vh∈Vh,t∈J,(52)phu~x,0=Πhpx,0,ph,tu~x,0=Πhptx,0,∀x∈Ω,(53)yh,ttu~,wh+bdivphu~,wh=bf,wh+bu~,wh,∀wh∈Wh,t∈J,(54)yhu~x,0=Phy0x,yh,tu~x,0=Phy1x,∀x∈Ω,(55)A−1qh,ttu~,vh+bdivqhu~,divvh+bzhu~,divvh=−phu~−pd,vh,∀vh∈Vh,t∈J,(56)qhu~x,T=0,qh,tu~x,T=0,∀x∈Ω,(57)zh,ttu~,wh=−yhu~−yd,wh,∀wh∈Wh,t∈J,(58)zhu~x,T=0,zh,tu~x,T=0,∀x∈Ω.It is clear that the exact solution and its approximation can be written in the following way:(59)p,y,q,z=pu,yu,qu,zu,ph,yh,qh,zh=phuh,yhuh,qhuh,zhuh.Lemma 1.
Letp,y,q,z be the solution of (19)–(26) and phu,yhu,qhu,zhu be the solution of (51)–(58) with u~=u, respectively. If the solution satisfies
(60)p,q∈L∞H12∩L2H22∩H2H12andy,z∈L∞H1,then we have
(61)y−yhuL∞L2+p−phuL∞L2≤Ch,z−zhuL∞L2+q−qhuL∞L2≤Ch,divp−phuL2L2+divq−qhuL2L2≤Ch.Proof.
For ease of presentation, we setρp=Πhp−phu, ρy=Phy−yhu, ρq=Πhq−qhu, ρz=Phz−zhu. From Equations (19)–(26) and (51)–(58) and the fact that divVh⊂Wh, with the definition of Ph and Πh, we can easily obtain the following error equations:
(62)A−1ρp,tt,vh+bdivρp,divvh=−bdivp−Πhp,divvh−A−1ptt−Πhptt,vh,∀vh∈Vh,(63)ρy,tt,wh+bdivρp,wh=−bdivp−Πhp,wh−ytt−Phytt,wh,∀wh∈Wh,(64)A−1ρq,tt,vh+bdivρp,divvh+bdivq−Πhq,divvh+bz−Phz,divvh=−A−1qtt−Πhqtt,vh−p−Πhp,vh−ρp,vh−bρz,divvh,∀vh∈Vh,(65)ρz,tt,wh=−ρy,wh−ztt−Phztt,wh,∀wh∈Wh.
Settingvh=ρp,t in (62), we have
(66)12ddtA−12ρp,t2+b12divρp2=−bdivp−Πhp,divρp,t−A−1ptt−Πhptt,ρp,t.
Notice that(67)bdivp−Πhp,divρp,t=−bdivp−Πhpt,divρp.
Substitute (67) into (66), integrating the resulting equation from 0 to t, and using Hölder’s inequality, Young’s inequality, Gronwall’s inequality, and the assumption on A and (34) and (35), we have
(68)ρp,tL∞L2+divρpL∞L2≤Ch.
Lettingwh=ρy,t in (63) and wh=−ρz,t in (65), respectively. Then integrating the resulting equations from 0 to t and t to T, respectively, we get
(69)ρy,tL∞L2≤CdivρpL2L2+Ch,ρz,tL∞L2≤CρyL2L2+Ch,where we also used Hölder’s inequality, Young’s inequality, and Gronwall’s inequality.
At last, settingvh=−ρq,t as the test function in (64) and integrating the resulting equation from t to T, similar to (68), we arrive at
(70)ρq,tL∞L2+divρpL∞L2≤Ch+CρpL2L2+ρzL2L2.
Note thatρp0=ρy0=ρqT=ρzT=0, and we have
(71)ρpL∞L2≤Cρp,tL2L2,(72)ρyL∞L2≤Cρy,tL2L2,(73)ρqL∞L2≤Cρq,tL2L2,(74)ρzL∞L2≤Cρz,tL2L2.
Combining (68)–(74), (32), (34), and (35) and the triangle inequality, we complete the proof.Using the same estimates as in Lemma1, we get the following.Lemma 2.
Letph,yh,qh,zh and phu,yhu,qhu,zhu be the solutions of (51)–(58) with u~=uh and u~=u, respectively. Then we have
(75)yh−yhuL∞L2+ph−phuL∞L2≤Cu−uhL2L2,zh−zhuL∞L2+qh−qhuL∞L2≤Cu−uhL2L2,divph−phuL2L2+divqh−qhuL2L2≤Cu−uhL2L2.Now, from the above Lemmas1 and 2, we can derive the following convergence results.Theorem 3.
Letu be the solution of (19)–(27) and uh be the solution of (41)–(49), respectively. Assume that all the assumptions in Lemma 1 are valid. Then we have
(76)u−uhL2L2≤Ch,(77)y−yhL∞L2+p−phL∞L2≤Ch,(78)z−zhL∞L2+q−qhL∞L2≤Ch,(79)divp−phL2L2+divq−qhL2L2≤Ch.Proof.
Letwh=1 in (31), and we have
(80)∫ΩPhudx=∫Ωudx.
By (80), we have
(81)∫0T∫ΩPhudxdt=∫0T∫Ωudxdt≥0.
Thus, we know thatPhu∈Kh.
It follows from (27) and (49) that
(82)u−uhL2L22=∫0Tu−uh,u−uhdt=∫0Tu−bz−bdivq,u−uhdt+∫0Tbz−bzhu+bdivq−qhu,u−uhdt+∫0Tbzhu−bzh+bdivqhu−qh,u−uhdt+∫0Tbzh+bdivqh−uh,u−uhdt≤∫0Tbz−bzhu+bdivq−qhu,u−uhdt−∫0Tu−bz−bdivq,u−Phudt+∫0Tbzhu−bzh+bdivqhu−qh,u−uhdt+∫0Tu−uh,u−Phudt+∫0Tbzh−bzhu+bdivqh−qhu,u−Phudt+∫0Tbzhu−bz+bdivqhu−q,u−Phudt≔∑i=16Ii.
Next, we estimate (82) term by term. For I1, using Hölder’s inequality, Young’s inequality, and Lemma 1, we have
(83)I1≤Ch2+14u−uhL2L22.
From (28), we find that
(84)u−bz−bdivq=const.
Thus(85)I2=−const∫0T∫Ωu−Phudxdt=0.
Setηp=phu−ph, ηy=yhu−yh, ηq=qhu−qh, and ηz=zhu−zh; then from (51)–(58), we have the following error equations:
(86)A−1ηp,tt,vh+bdivηp,divvh=bu−buh,divvh,∀vh∈Vh,t∈J,(87)ηy,tt,wh+bdivηp,wh=bu−buh,wh,∀wh∈Wh,t∈J,(88)A−1ηq,tt,vh+bdivηq,divvh=−ηp,vh−bηz,divvh,∀vh∈Vh,t∈J,(89)ηz,tt,wh=−ηy,wh,∀wh∈Wh,t∈J.choosing vh=ηq in (86), wh=ηz in (87), vh=ηp in (88), and wh=ηy in (89), respectively. Since ηp0=ηy0=ηp,t0=ηy,t0=ηqT=ηzT=ηq,tT=ηz,tT=0, integrating the resulting equations from 0 to T, we can see that
(90)I3=∫0Tηz+divηq,bu−buhdt=−ηp2−ηy2≤0.
ForI4 and I5, using Hölder’s inequality, Young’s inequality, Lemma 2, and (32), we get
(91)I4≤Ch2+14u−uhL2L22,(92)I5≤ηzL2L2+divηqL2L2u−PhuL2L2≤u−uhL2L2u−PhuL2L2≤Ch2+14u−uhL2L22.
ForI6, by Hölder’s inequality, the triangle inequality, Lemma 1, and (32), we arrive at
(93)I6≤b0,∞zhu−zL2L2+divηqL2L2u−PhuL2L2≤Ch2.
Combining (82), (83), (85), and (90)–(92), we derive (76). Using (76), Lemmas 1 and 2, and the triangle inequality, we complete the proof.


## 4. Superconvergence Properties

In this section, we will derive some superconvergence properties for the optimal control problems. In order to derive the main results, we need the following lemmas.Lemma 4.
Letp,y,q,z be the solution of (19)–(27) and phu,yhu,qhu,zhu be the solution of (51)–(58) with u~=u, respectively. If the solution satisfies
(94)p,q∈L∞H12∩L2H22∩H1H22andy,z∈L∞H1,then we have
(95)Phy−yhuL∞L2+Πhp−phuL∞L2≤Ch32,(96)Phz−zhuL∞L2+Πhq−qhuL∞L2≤Ch32,(97)divΠhp−phuL2L2+divΠhq−qhuL2L2≤Ch32.Proof.
At first, for anyp∈V and v∈Vh, by applying the proof of Theorems 4.1, 5.1, and Example 6.2 in [32], we can prove
(98)A−1p−Πhp,vh≤Ch32p2vh+divvh.
Moreover, using (32) and (35), we have
(99)bdivp−Πhp,divvh=b−Phbdivp−Πhp,divvh≤Ch2b1,∞p2divvh,by−Phy,divvh=b−Phby−Phy,divvh≤Ch2b1,∞y1divvh.
Similar to Lemma1, using (98), we can prove (95)–(97). We omit the proof here.Lemma 5.
LetphPhu,yhPhu,qhPhu,zhPhu and phu,yhu,qhu,zhu be the solutions of (51)–(58) with u~=Phu and u~=u, respectively. Then we have
(100)yhu−yhPhuL∞L2+phu−phPhuL∞L2=0,zhu−zhPhuL∞L2+qhu−qhPhuL∞L2=0,divphu−phPhuL2L2+divqhu−qhPhuL2L2=0.Proof.
First, setζp=phu−phPhu, ζy=yhu−yhPhu, ζq=qhu−qhPhu, and ζz=zhu−zhPhu, and we choose u~=Phu and u~=u in (51)–(58), respectively; then we obtain the following error equations:
(101)A−1ζp,tt,vh+bdivζp,divvh=bu−bPhu,divvh,∀vh∈Vh,t∈J,(102)ζy,tt,wh+bdivζp,wh=bu−bPhu,wh,∀wh∈Wh,t∈J,(103)A−1ζq,tt,vh+bdivζq,divvh=−ζp,vh−bζz,divvh,∀vh∈Vh,t∈J,(104)ζz,tt,wh=−ζy,wh,∀wh∈Wh,t∈J.
Noting from the fact thatdivVh⊂Wh and (31)
(105)bu−bPhu,divvh≤Ch2b1,∞u1divvh,∀vh∈Vh,bu−bPhu,wh≤Ch2h2b1,∞u1wh,∀wh∈Wh,then, as in Lemma 1, using the stability estimates, we complete the proof.Lemma 6.
LetphPhu,yhPhu,qhPhu,zhPhu and ph,yh,qh,zh be the solutions of (51)–(58) with u~=Phu and u~=uh, respectively. Then we have
(106)∫0TbzhPhu−bzh,Phu−uh+bdivqhPhu−qh,Phu−uhdt≤0.Proof.
Setθp=ph−phPhu, θy=yh−yhPhu, θq=qh−qhPhu, and θz=zh−zhPhu, similar to (101)–(104), and we obtain the following error equations:
(107)A−1θp,tt,vh+bdivθp,divvh=buh−bPhu,divvh,∀vh∈Vh,(108)θy,tt,wh+bdivθp,wh=buh−bPhu,wh,∀wh∈Wh,(109)A−1θq,tt,vh+bdivθq,divvh=−θp,vh+bθz,divvh,∀vh∈Vh,(110)θz,tt,wh=−θy,wh,∀wh∈Wh,choosing vh=θq in (107), wh=θz in (108), vh=−θp in (109), and wh=−θy in (110), respectively. Integrating the resulting equations from 0 to T, we derive
(111)∫0TzhPhu−zh,bPhu−buh+divqhPhu−qh,bPhu−buhdt=−θyL2L22−θpL2L22,which yields to (106).Theorem 7.
Letu be the solution of (19)–(27) and uh be the solution of (41)–(49), respectively. Assume that all the assumptions in Lemma 1 are valid. Then, we have
(112)Phu−uhL2L2≤Ch32.Proof.
We chooseu~=uh in (23) and u~h=Phu in (44) to get the following two inequalities:
(113)∫0Tu−bz−bdivq,uh−udt≥0,∫0Tuh−bzh−bdivqh,Phu−uhdt≥0.
Note thatuh−u=uh−Phu+Phu−u. Adding the above two inequalities, we get
(114)∫0Tuh−u+bz−bzh+bdivq−qh,Phu−uhdt+∫0Tu−bz−bdivq,Phu−udt≥0.
Thus, by (114) and (26), we find that
(115)Phu−uhL2L22=∫0TPhu−u,Phu−uhdt+∫0Tu−uh,Phu−uhdt≤∫0Tbz−bzh+bdivq−bdivqh,Phu−uhdt+∫0Tu−bz−bdivq,Phu−udt=∫0Tbz−bPhz+bdivq−bdivΠhq,Phu−uhdt+∫0TbPhz−bzhu+bdivΠhq−divqhu,Phu−uhdt+∫0Tbzhu−bzhPhu+bdivqhu−bdivqhPhu,Phu−uhdt+∫0TbzhPhu−bzh+bdivqPhu−bdivqh,Phu−uhdt+∫0Tu−bz−bdivq,Phu−udt≔∑i=15Ji.
At first, from Cauchy inequality, (26), (28), and (24), it is easy to get
(116)J1=∫0Tbz−bPhz+bdivq−bdivΠhq,Phu−uhdt≤Ch4+14Phu−uhL2L22.
Using Hölder’s inequality, Young’s inequality, and Lemmas4 and 5, we arrive at
(117)J2≤Ch3+14Phu−uhL2L22,(118)J3≤Ch3+14Phu−uhL2L22.
Combining (84), (85), and (115)–(118) and Lemma 6, we derive (112).Using Theorem7 and the stability estimates as in Lemma 1, we can arrive at the following.Lemma 8.
LetphPhu,yhPhu,qhPhu,zhPhu and ph,yh,qh,zh be the solutions of (51)–(58) with u~=Phu and u~=uh, respectively. Assume that all the assumptions in Theorem 7 are valid. Then we have
(119)yh−yhPhuL∞L2+ph−phPhuL∞L2≤Ch32,zh−zhPhuL∞L2+qh−qhPhuL∞L2≤Ch32,divph−phPhuL2L2+divqh−qhPhuL2L2≤Ch32.Combining Lemma4–8 and using the triangle inequality, we derive the following superconvergence results.Theorem 9.
Lety,p,z,q and yh,ph,zh,qh be the solutions of (19)–(27) and (41)–(49), respectively. Assume that all the assumptions in Theorem 7 are valid. Then we have
(120)Phy−yhL∞L2+Πhp−phL∞L2≤Ch32,Phz−zhL∞L2+Πhq−qhL∞L2≤Ch32,divΠhp−phL2L2+divΠhq−qhL2L2≤Ch32.


## 5. Numerical Experiments

In this section, we present a numerical example to validate our convergence and superconvergence results. The hyperbolic OCP was dealt numerically with codes developed based on AFEPack. The package is freely available and the details can be found in [33].Letτ>0,N=T/τ∈ℤ, and tn=nτ,n=0,1,⋯,N. Set
(121)ϕn=ϕx,tn,ϕttn=ϕn+1−2ϕn+ϕn−1/τ2.Then, a fully discrete splitting positive definite mixed finite element solutionphn,yhn,qhn,zhn,uhn of (1)–(6) satisfies the following system:
(122)A−1ph,ttn,vh+divphn,divvh=fn,divvh+uhn,divvh,∀vh∈Vh,n=1,2,⋯,N.ph1−ph−12τ,vh=ptx,0,vh,∀vh∈Vh,ph0=Πhpx,0,∀x∈Ω,yh,ttn,wh+divphn,wh=fn,wh+uhn,wh,∀wh∈Wh,n=1,2,⋯,N,yh1−yh−12τ,wh=y1x,wh,∀wh∈Wh,yh0=Phy0x,∀x∈Ω,A−1qh,ttn,vh−divqhn,divvh=phn−pdn,vh+zhn,divvh,∀vh∈Vh,n=N,N−1,⋯,1,qhN−1−qhN+12τ,vh=qtx,T,vh,∀vh∈Vh,qhN=0,∀x∈Ω,zh,ttn,wh=yhn−ydn,wh,∀wh∈Wh,n=N,N−1,⋯,1,zhN−1−zhN+12τ,wh=ztx,T,wh,∀wh∈Wh,zhN=0,∀x∈Ω,uhn=max0,−zh+divqh¯+zhn+divqhn,∀n=0,1,⋯,N.Example 10.
LetΩ=0,1×0,1,T=1,cx=1,A=E, where E denotes identity matrix. The data under testing are as follows:
(123)yx,t=t2sin2πx1sin2πx2,px,t=−2πt2cos2πx1sin2πx22πt2sin2πx1cos2πx2,qx,t=1−t2cos2πx1sin2πx21−t2sin2πx1cos2πx2,zx,t=−divqx,t+21−tsin2πx1sin2πx2,ydx,t=yx,t−zttx,t,pdx,t=px,t−qttx,t+2π1−tcos2πx1sin2πx22π1−tsin2πx1cos2πx2,ux,t=max0,−zx,t+divqx,t¯+zx,t+divqx,t,fx,t=yttx,t+divpx,t−ux,t.Leteu=u−uh,ey=y−yh,ep=p−ph,ez=z−zh,eq=q−qh and Eu=Phu−uh,Ey=Phy−yh,Ep=Πhp−ph,Ez=Phz−zh,Eq=Πhq−qh. We set ·L∞L2=·∞,2 and ·L2L2=·2,2. The numerical results of convergence and superconvergence based on a sequence of uniformly refined meshes are reported in Tables 1–4, respectively. They are consistent with our theoretical results.Table 1
Numerical results of convergence.
h=τeu2,2Rateey∞,2Rateez∞,2Rate1/104.9561e−2—3.5081e−2—4.0448e−2—1/202.5280e−20.971.8105e−20.952.0416e−20.991/401.2642e−21.009.1468e−30.991.0208e−21.001/806.2663e−31.014.5615e−31.005.1032e−31.00Table 2
Numerical results of convergence.
h=τep∞,2Rateeq∞,2Ratedivep2,2Ratediveq2,2Rate1/104.7647e−2—4.9557e−2—5.5145e−2—6.7192e−2—1/202.4324e−200.972.5128e−20.982.8071e−20.973.3987e−20.981/401.2153e−21.001.28077e−20.971.4045e−21.001.6943e−21.001/806.0748e−31.006.4102e−31.007.0231e−31.008.4158e−31.01Table 3
Numerical results of superconvergence.
hτEu2,2RateEy∞,2RateEz∞,2Rate1/101/104.8542e−2—3.4063e−2—4.0284e−2—1/201/301.7282e−21.491.2154e−21.491.4404e−21.481/401/906.0473e−31.514.3104e−31.505.0486e−31.511/801/2702.1158e−31.521.4835e−31.541.7635e−31.52Table 4
Numerical results of superconvergence.
hτEp∞,2RateEq∞,2RatedivEp2,2RatedivEq2,2Rate1/101104.6684e−2—4.7694e−2—5.4134e−2—6.5048e−2—1/201/301.6902e−21.471.7120e−21.481.9402e−21.482.3406e−21.471/401/905.9615e−31.506.0452e−31.506.8342e−31.518.1906e−31.511/801/2702.0512e−31.542.1124e−31.522.3448e−31.542.8302e−31.53



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*Source: 3520668-2022-01-06.xml*