As a byproduct, we show a k+ 2-th order superconvergence of the DG solution towards the Gauss-Radau projection of the exact solution

SUPERCONVERGENCE OF DISCONTINUOUS GALERKIN METHOD FOR SCALAR NONLINEAR HYPERBOLIC EQUATIONS

WAIXIANG CAO^∗, CHI-WANG SHU^†, YANG YANG^‡, AND ZHIMIN ZHANG^{§ ¶}

Abstract. In this paper, we study the superconvergence behavior of the semi-discrete discon- tinuous Galerkin (DG) method forscalarnonlinear hyperbolic equations in one spatial dimension.

Superconvergence results for problems with fixed and alternating wind directions are established.

On the one hand, we prove that, if the wind direction is fixed (i.e., the derivative of the flux function is bounded away from zero), both the cell average error and numerical flux error at cell interfaces converge at a rate of 2k+ 1 when upwind fluxes and piecewise polynomials of degreekare used.

Moreover, we also prove that the function value approximation of the DG solution is superconver- gent at interior right Radau points, and the derivative value approximation is superconvergent at interior left Radau points, with an order ofk+ 2 andk+ 1, respectively. As a byproduct, we show a k+ 2-th order superconvergence of the DG solution towards the Gauss-Radau projection of the exact solution. On the other hand, superconvergence results for problems with alternating wind directions (i.e., the derivative of the flux function either changes sign or otherwise achieves the value zero in the domain) are also established. To be more precise, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an order ofk+ 2, when Godunov fluxes are used. We then prove that the highest superconvergence rate of the DG solution itself isk+³₂ when sonic points (i.e., the derivative of the flux function achieves zero) appear in the computational domain. As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell average. Numerical experiments demonstrate that most of our results are optimal, i.e., the superconvergence rates are sharp.

Key words.Discontinuous Galerkin methods, superconvergence, nonlinear hyperbolic equations

AMS subject classifications. 65M15, 65M60, 65N30

1. Introduction. In this paper, we investigate the superconvergence behavior of the discontinuous Galerkin (DG) method for the following one-dimensional nonlinear hyperbolic equation

ut+f(u)x= 0, (x, t)∈[a, b]×[0, T], u(x,0) =u₀(x), x∈Ω = [a, b],¯ (1)

where u₀ is sufficiently smooth. We assume that the nonlinear flux functionf(u) is sufficiently smooth with respect touand the final timeT is not too large so that the exact solutions are smooth. In this paper, for simplicity, we only consider the periodic boundary condition.

The superconvergence behavior of the DG (see, e.g., [12, 13, 14, 15, 16, 17]) method has been studied for many years, and has been a hot research topic in recent

∗School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China. Email:

caowx@bnu.edu.cn. Research was supported in part by NSFC grant No.11501026, and the China Postdoctoral Science Foundation grants 2016T90027 and 2015M570026.

†Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail:

shu@dam.brown.edu. Research was supported in part by DOE grant DE-FG02-08ER25863 and NSF grant DMS-1418750.

‡Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA. E-mail: yyang7@mtu.edu. Research was supported in part by NSFC grants No.11571367 and No.11601536 and Michigan Technological University Research Excellence Fund Scholarship and Creativity Grant 1605052

§Beijing Computational Science Research Center, Beijing 100193, China. E-mail:

zmzhang@csrc.ac.cn. Research was supported in part by NSFC grants No.11471031 and No.91430216.

¶ Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. E-mail:

zzhang@math.wayne.edu. Research was supported in part by NSF grant DMS-1419040.

1

years. We refer to [1,2,3,10,11,18,23,24,4,19] for an incomplete list of references on the superconvergence of DG methods for hyperbolic problems. However, all the studies in the literature are based on linear equations. Only recently, Meng et al.

studied the superconvergence of semi-discrete DG methods for (1) with fixed wind direction, i.e., |f^′(u)| possesses a uniform positive lower bound [20]. They proved that the order of the error between the DG solution and a particular projection of the exact solution can achievek+³₂ when upwind fluxes were used. Compared with the linear case, where the highest superconvergence rate can reach 2k+ 1 (see, e.g., [7, 6]), the superconvergence rate k+ ³₂ is far from optimal. Furthermore, to our best knowledge, no superconvergence results of DG methods applied to nonlinear hyperbolic equations with sonic points (i.e., points at whichf^′(u) = 0) are available in the literature.

The main purpose of this paper is to study superconvergence properties of DG methods for (1) with general flux functions. If the wind direction is fixed, our analy- sis indicates a 2k+ 1-th order convergence rate of the numerical flux at mesh nodes and for the cell average, and a k+ 2-th order of the error between the DG solution and the Gauss-Radau projection of the exact solution as well as the function value error at interior right Radau points, and a k+ 1-th order of the derivative error at interior left Radau points. As we may recall, these superconvergence results are the same as for non-degenerate linear hyperbolic problems (see, [7,6]). While if the wind direction is changing in the computational domain, we proved that the superconver- gence phenomenon still exists and the convergence rate may depend upon the specific property of the flux functions. Specifically, we first prove that the DG flux function is superconvergent towards a particular flux function of the exact solution, with an error boundO(h^k+2), when Godunov fluxes are used. Then we establish the supercloseness result between the DG solution itself and a particular projection of the exact solution, and reveal that the highest superconvergence rate for nonlinear hyperbolic equations with sonic points isO(h^k+32). As byproducts, we obtain superconvergence properties for the DG solution and the DG flux function at special points and for cell averages.

These superconvergence results are similar to linear problems with degenerate variable coefficients, see [5].

The contribution of this paper is to provide a symmetric method to study the superconvergence behavior of the DG methods for nonlinear problems. On the one hand, we establish the superconvergence results for the monotone flux, and prove that all the superconvergence results for the linear case in [7, 6] still hold true for nonlinear problems. Furthermore, the superconvergence results established in this paper improve those of [20] to the possible optimal superconvergence rates. On the other hand, we uncover the superconvergence phenomenon of the DG methods for (1) with the degenerate flux function, and extend the superconvergence result for linear problems to a more general nonlinear case. By doing so, we present a full picture for the superconvergence properties of DG methods applied to the (degenerate) nonlinear hyperbolic problems, and enrich the superconvergence theory of the DG method for linear hyperbolic equations in one dimension.

To end the introduction, we would like to emphasize that the main difficulty in the superconvergence analysis for nonlinear problems is how to deal with the nonlinear terms in the error equation. Our analysis is along the following lines: we first use a Taylor expansion to linearize the error equation; subsequently, we make an a priori error assumption to deal with the nonlinearity of the flux and other high-order terms in the linearization, then the superconvergence analysis for nonlinear problems is reduced to a linear one; finally, we introduce a correction function to deal with the

2

linear part to obtain higher-order accuracy.

The rest of the paper is organized as follows. In Section 2, we present DG schemes for the one-dimensional nonlinear hyperbolic conservation laws, and discuss the choice of numerical fluxes. Section 3 is dedicated to the superconvergence analysis of the DG methods for problems with fixed wind direction. Superconvergence results are established by using the idea of correction function, Taylor expansion, and an a priori error assumption. In Section 4, we study the superconvergence behavior of the DG approximation for problems with sonic points, and prove that the DG flux function is superconvergent with an order ofO(h^k+2) to a particular flux function of the exact solution. Superconvergence behavior of the DG solution itself is also discussed in this case. We reveal a very important fact that the superconvergence phenomena are also valid for general fluxes, and the superconvergence rates may depend upon specific properties of the flux functions. In Section 5, we provide numerical examples to support our theoretical findings. Finally, concluding remarks and some possible future works are presented in Section 6.

Throughout this paper, we adopt standard notations for Sobolev spaces such as W^m,p(D) on sub-domainD ⊂Ω equipped with the norm∥ · ∥m,p,D and semi-norm

|·|m,p,D. WhenD= Ω, we omit the indexD; and ifp= 2, we setW^m,p(D) =H^m(D),

∥ · ∥m,p,D=∥ · ∥m,D, and| · |m,p,D=| · |m,D. NotationA≲B implies thatA can be bounded byB multiplied by a constant independent of the mesh sizeh.

2. DG schemes and the energy inequality. Let Ω = [0,2π] and 0 =x1 2 <

x³

2 < . . . < x_N+¹

2 beN+ 1 distinct points on the interval Ω. For all positive integers r, we defineZr={1, . . . , r}, and denote by

τj= (x_j−1 2, x_j+1

2), xj =1 2(x_j−1

2 +x_j+1

2), j∈ZN

the cells and cell centers, respectively. Let h_j =x_j+1

2 −x_j−1

2, h_j = h_j/2 and h= max

j h_j. We assume that the mesh is regular, i.e., the ratio between the maximum and minimum mesh sizes shall stay bounded during mesh refinements.

Define the discontinuous finite element space

V_h={v: v|τj ∈Pk(τ_j), j∈ZN},

where Pk denotes the space of polynomials of degree at most k with coefficients as functions of t. The DG scheme for (1) reads as: Find uh ∈ Vh such that for any v∈Vh

(2) ((uh)t, v)j−(f(uh), vx)j+ ˆf(uh)v⁻

j+¹₂ −fˆ(uh)v⁺

j−¹₂ = 0, j∈ZN, where (u, v)_j =∫

τjuvdx,v⁻

j+¹₂ andv⁺

j+¹₂ separately denote the left and right limits of vat the pointx_j+1

2, and ˆf(uh) is the numerical flux, which is a single-valued function defined at each cell interface and in general depends on the values of the numerical solutionuh from both sides of the interface, i.e.,

fˆ(uh)

j+¹₂ = ˆf(uh(x⁻_j+1 2

), uh(x⁺_j+1 2

)), ∀j∈ZN.

The choice of the numerical flux is of great significance in assuring the stability of the DG scheme. Iff^′(u)≥δ >0, for all x∈Ω, we define the numerical flux as

fˆ(uh) =f(u⁻_h).

3

Similarly, iff^′(u)≤ −δ <0, we choose

fˆ(u_h) =f(u⁺_h).

However, iff^′(u) changes its sign in the computational domain, we take the Godunov flux as our numerical flux, i.e.,

fˆ(uh) = ˆf(u⁻_h, u⁺_h) =

{ min_u−

h≤µ≤u⁺_h f(µ), if u⁻_h < u⁺_h, max_u+

h≤µ≤u⁻_h f(µ), if u⁻_h ≥u⁺_h. (3)

Note that the upwind flux for problems with alternating wind direction is defined as follows.

f(u˜ _h) =

{ f(u⁻_h), if f^′(u)>0, f(u⁺_h) if f^′(u)≤0.

(4)

The above flux is an essential part in our later energy estimate and error analysis for problems with alternating wind direction.

Throughout this paper, we will use the following notation (5) e=u−uh, ξ=uI−uh, η=u−uI. Hereu_I is some special interpolation function ofuto be determined.

To end with this section, we would like to present the following energy inequality, which will be frequently used in our later superconvergence analysis.

Theorem 1. Let ube the solution of (1)anduI ∈Vh be some special interpola- tion function of u. Assume that uh is the solution of (2) with the numerical fluxes chosen as the upwind flux (for fixed wind directions) or Godunov flux (for alternating wind directions) . Then for both numerical fluxes,

(6) 1 2

d

dt∥ξ∥²0≲h⁻²∥e∥²0,∞∥η∥²0+ (1 +h⁻¹∥e∥0,∞)∥ξ∥²0+h⁻¹∥e∥²0,∞

∑N j=1

(˜η_j+1

2)²+|I|+|J|, whereη˜ is the numerical flux defined in (4)with f(u_h)replaced byη, and

I=

∑N j=1

( ˜f(uh)−fˆ(uh))[ξ]

j+¹₂, (7)

J =−(ηt, ξ) + (f^′(u)η, ξx) +

∑N j=1

(f^′(u)˜η[ξ])

j+¹₂. (8)

Proof. Since the exact solution also satisfies (2), we have for allv∈Vh

(9) (et, v)j−(f(u)−f(uh), vx)j+ (f(u)−fˆ(uh))v⁻|j+¹₂−(f(u)−fˆ(uh))v⁺|j−¹2 = 0.

Summing up overjfrom 1 toN and using the periodic boundary condition, we obtain (et, v) = (f(u)−f(uh), vx) +

∑N j=1

(

(f(u)−fˆ(uh))[v])

j+¹₂,

4

where (u, v) =∑N

j=1(u, v)j and [v]_j+1

2 =v⁺_j+1

2 −v_j+⁻ 1 2

denotes the jump of v at the pointx_j+1

2. Now we takev=ξin the above identity to get (10) (ξt, ξ) =−(ηt, ξ) + (f(u)−f(uh), ξx) +

∑N j=1

(

(f(u)−f˜(uh))[ξ])

j+¹₂ +I.

By the Taylor expansion off aboutu, we have

f(u)−f(uh) =f^′(u)ξ+f^′(u)η−1 2

f¯_u^′′(ξ+η)², f(u)−f(u˜ h) =f^′(u) ˜ξ+f^′(u)˜η−1

2

¯¯

f^′′_u( ˜ξ+ ˜η)², (11)

where ¯f_u^′′=f^′′(α1u+ (1−α1)uh) and ¯f¯^′′_u=f^′′(α2u+ (1−α2)uh) with 0≤α1, α2≤1.

Then (10) can be rewritten as

(12) (ξ_t, ξ) =I₁−I2

2 +I+J, where

I1= (f^′(u)ξ, ξx) +

∑N j=1

(

f^′(u) ˜ξ[ξ])

j+¹₂, I2= ( ¯f_u^′′(ξ+η)², ξx) +

∑N j=1

(f¯¯_u^′′( ˜ξ+ ˜η)²[ξ])

j+¹₂. Let

(13) Ω₁={j∈ZN :f_j+^′ 1

2 ≥0}, Ω₂={j∈ZN :f_j+^′ 1 2

<0}, wheref_j+^′ 1

2

=f^′(u(x_j+1

2, t)). By a simple integration by parts, we have

I1=−1

2(∂xf^′(u), ξ²)−

∑N j=1

(f^′(u){ξ}[ξ])|j+¹₂ + ∑

j∈Ω1

(f^′(u)ξ⁻[ξ])|j+¹₂ + ∑

j∈Ω2

(f^′(u)ξ⁺[ξ])|j+¹₂

=−1

2(∂xf^′(u), ξ²)− ∑

j∈Ω1

(f^′(u)[ξ]²)|j+¹₂ + ∑

j∈Ω2

(f^′(u)[ξ]²)|j+¹₂ ≤ −1

2(∂xf^′(u), ξ²)≤C∥ξ∥²0. Here and in the following, C is a positive constant independent of the mesh sizeh,

and is not necessary to be the same at every occurrence. As for I2, we have from Cauchy-Schwarz inequality

|I2| ≤C∥e∥0,∞



∥e∥0∥ξx∥0+∥ξ∥²Γh+

∑N j=1

η˜_j+1 2[ξ]_j+1

2





≲h⁻¹∥e∥0,∞(

∥e∥0∥ξ∥0+∥ξ∥²0

)+∥e∥0,∞

∑N j=1

˜η_j+1 2[ξ]_j+1

2

≲h⁻²∥e∥²0,∞∥η∥²0+ (1 +h⁻¹∥e∥0,∞)∥ξ∥²0+h⁻¹∥e∥²0,∞

∑N j=1

(˜η_j+1 2)²,

5

where for any givenv,

∥v∥Γh =



∑^N

j=1

(v⁻

j+¹₂)²+ (v⁺

j+¹₂)²





1 2

,

and in the second and last steps, we have used the inverse inequality

∥v_x∥0≤Ch⁻¹∥v∥0, ∥v∥Γh ≤Ch⁻¹²∥v∥0, ∥v∥0,∞,τj ≤Ch⁻¹²∥v∥0,τj, ∀v∈V_h. Then (6) follows by substituting the estimates of I₁ and I₂ into (12). The proof is complete.

3. Analysis for problems with fixed wind direction. In this section, we study the superconvergence of DG methods for problems with fixed wind direction.

Without loss of generality, we consider the case f^′(u)≥δ >0. The same arguments can be applied to the case withf^′(u)≤ −δ <0.

To study the superconvergence properties of the DG solution, our analysis is along this line: We first construct a special interpolation function uI ∈ Vh of the exact solution such that the DG solutionuh is super-close touI underL² norm; and then we analyze the superconvergence behavior of the interpolation function uI; finally, the superconvergence for the DG solution is reduced to the superconvergence of the interpolation function uI due to the super-closeness betweenuh and uI. Therefore, our goal here is to design the special functionuI.

Note that

∥ξ(·, t)∥²0=∥ξ(·,0)∥²0+

∫ t 0

d

dt∥ξ∥²0dt, ∀t∈(0, T],

then the convergence rate of∥ξ∥0 depends upon the term _dt^d∥ξ∥²0. In light of (6) and the fact that I in (6) vanishes for fixed wind directions, to achieve our superconver- gence goal, u_I should be specially designed such that J in (6) is of high order. To this end, we begin with some preliminaries.

3.1. Preliminaries. First, we denote byLn andLj,n the traditional Legendre polynomials of degreenin the intervals [−1,1] andτj, j∈ZN, respectively. For any x∈τj, let

(14) ω(x, t) =f^′(u(x, t)), w¯j(s, t) =ω(xj+ ¯hjs, t) =ω(x, t), s∈[−1,1].

Note that for any fixed t and j, where j ∈ ZN, ¯ωj ≥δ > 0 ∈ L¹(τ0), τ0 = [−1,1], there exists a series of monic orthogonal polynomials{ϕ¯j,n}^∞n=1 with respect to the weight function ¯ωj, i.e.,

(15) ( ¯ϕj,n,ϕ¯j,m)ω¯_j :=

∫ 1

−1

¯

ωjϕ¯j,nϕ¯j,mds=γnδmn,

where γn = ( ¯ϕj,n,ϕ¯j,n)ω¯_j and δmn is the Kronecker delta. Moreover, ¯ϕj,n can be constructed by the following three-term recurrence formula

(16) ϕ¯j,0= 1, ϕ¯j,1=s−α0, ϕ¯j,n+1= (s−αn) ¯ϕj,n−βnϕ¯j,n−1, n≥1, where for alln≥0, m≥1,

(17) αn = (sϕ¯j,n,ϕ¯j,n)ω¯_j/( ¯ϕj,n,ϕ¯j,n)ω¯_j, βm= ( ¯ϕj,m,ϕ¯j,m)ω¯_j/( ¯ϕj,m−1ϕ¯j,m−1)ω¯_j.

6

With the orthogonal polynomials{ϕ¯j,n}on [−1,1], we can construct a set of orthog- onal polynomialsϕj,n in each elementτj as follows.

ϕ_j,n(x, t) = ¯ϕ_j,n(s, t) = ¯ϕ_j,n(2(x−xj) h_j , t)

, x∈τ_j.

Moreover, there holds the following orthogonal property (18)

∫

τ_j

(ϕ_j,nϕ_j,mω)(x, t)dx= h_j 2

∫ 1

−1

( ¯ϕ_j,nϕ¯_j,mω¯_j)(s, t)ds= h_j 2 γ_nδ_mn.

That is, {ϕj,n} are orthogonal polynomials with respect to the weight functionω = f^′(u) in τj.

Lemma 2. For any j ∈ ZN, let {ϕ_j,n} be a sequence of orthogonal polynomials with respect to the weight functionω=f^′(u)inτj, andf^′(u)be a sufficiently smooth function satisfying|∂_t^mf^′(u)|≲1, m≤k+ 1. Then

(19) ∥ϕ_j,n∥0,τj ≲h¹², ∥ϕ_j,n∥0,∞,τj ≲h⁻¹²∥ϕ_j,n∥0,τj ≲1.

Moreover, for all positivem, where1≤m≤k+ 1, there holds

(20) ∥∂_t^mϕ_j,n∥0,∞≲1.

Proof. First, (19) can be easily verified by a direct calculation. To show (20), it is sufficient to prove

(21) |∂^m_t ϕ¯_j,n|≲1, 1≤m≤k+ 1.

We will prove (21) by induction. For any integer m≥1, we have, from (16) and the Newton-Leibniz formula of derivative,

∂_t^mϕ¯j,1=−∂^m_t α0=−∂^m_t (∫1

−1sω¯jds

∫1

−1ω¯_jds )

≲

∑m

r=0∥∂^r_tω¯j∥0,∞

∫1

−1ω¯jds^m+1

≲1.

Then (21) holds for n= 1. Now we suppose (21) is valid for alln and prove it also holds forn+ 1. Actually, recall the three-term recurrence formula of ¯ϕj,n in (16), we easily get

|∂_t^mϕ¯j,n+1|≲(|∂_t^mαn|+|∂_t^mβn|) (|∂^m_t ϕ¯j,n|+|∂_t^mϕ¯j,n−1|)≲(|∂^m_t αn|+|∂_t^mβn|). Again, we use the Newton-Leibniz formula of derivative to obtain

(|∂_t^mαn|+|∂^m_t βn|)≲

∑m

r=0|∂_t^rϕ¯j,n| ( ¯ϕj,n,ϕ¯j,n)^m+1_ω¯j +

∑m

r=0(|∂_t^rϕ¯j,n|+|∂^r_tϕ¯j,n−1|) ( ¯ϕj,n−1,ϕ¯j,n−1)^m+1_ω¯j ≲1, and thus,

|∂^m_t ϕ¯j,n+1|≲1.

Then (21) is also valid forn+ 1, and this finishes our proof.

7

Let

H_h¹={v: v|τ_j ∈H¹(τj), j∈ZN},

and for allψ∈H_h¹, we denote byP_h⁻ψ∈Vh the traditional Gauss-Radau projection ofψ, which is defined by

(22) (P_h⁻ψ, v)j= (ψ, v)j, ∀v∈Pk−1, P_h⁻ψ(x⁻_j+1 2

) =ψ_j+⁻ 1 2

, j∈ZN.

Note that the Gauss-Radau projectionP_h⁻is frequently used in the DG error analysis.

Now we extend the definition of Gauss-Radau projection to a more general one. Given a positive functionω, we define a projectionPh:H_h¹→Vh by

(23)

∫

τ_j

ωP_hψvdx=

∫

τ_j

ωψvdx, ∀v∈Pk−1, P_hψ(x⁻

j+¹₂) =ψ(x⁻

j+¹₂).

Note thatPh is reduced to the traditional Gauss-Radau projectionP_h⁻ whenω = 1.

Furthermore, we have the following approximation properties for the projectionPh. Lemma 3. Let ω = f^′(u) ≥ δ > 0. Then the projection Ph in (23) is well- defined. Moreover, if ψ ∈ W^k+2,∞(Ω) and |∂_xⁿf^′(u)| ≲ 1, n ≤ k, there holds the following results.

• Phψ is super-close to the Gauss-Radau projectionP_h⁻ψ, i.e., (24) ∥∂_t^m(P_hψ−P_h⁻ψ)∥0,∞,τj ≲h^k+2∥∂^m_t ψ∥k+1,∞, m= 0,1.

• The function value approximation of Phψ is superconvergent at the interior right Radau pointsrj,m,1≤m≤k+ 1(zeros of the right Radau polynomial Lj,k+1−Lj,k except the pointx_j+¹

2), namely,

(25) (Phψ−ψ)(rj,m)≲h^k+2∥ψ∥k+2,∞, m∈Zk.

• The derivative value approximation ofP_hψis superconvergent at the interior left Radau points l_j,m, m∈Zk (zeros of left Radau polynomial L_j,k+1+L_j,k except the pointx=x_j−1

2 ), i.e.,

(26) ∂x(Phψ−ψ)(lj,m)≲h^k+1∥ψ∥k+2,∞, m∈Zk.

• The cell average of Phψ in each element τj is superconvergent with an order of 2k+ 1, i.e.,

(27)

1 hj

∫

τj

(ψ−Phψ)(x)dx

≲h^2k+1∥ψ∥k+1,τ_j. Proof. SincePhψ∈Vh, we express in each elementτj

P_hψ

τ_j =

∑k n=0

a_nϕ_j,n(x),

where {an}^k0 are constants to be determined. Recall the definition of Ph and the orthogonality of{ϕj,n} in (18), we easily obtain

an=

∫

τ_jωψϕj,ndx

∫

τjωϕ_j,nϕ_j,ndx = 2 hjγn

∫

τj

ωψϕj,ndx, n≤k−1,

8

whereγn is the same as in (15). Moreover, noticing that the zeros of the orthogonal polynomials{ϕ¯j,n}are all real, simple, and lies in the interval (−1,1) (see [22], Theo- rem 3.2), we have ¯ϕj,n(1)̸= 0, or equivalently,ϕj,n(x⁻_j+1

2

)̸= 0, which yields, together with the second identity of (23),

ak= 1 ϕ_j,k(x⁻_j+1

2

) (

ψ(x⁻_j+1 2

)−

k−1

∑

n=0

anϕj,n(x⁻_j+1 2

) )

= 1

ϕ¯j,k(1) (

ψ(x⁻_j+1 2

)−

k∑−1 n=0

anϕ¯j,n(1) )

.

Therefore, the projectionPhψfor any functionψ∈H_h¹ is uniquely determined.

Now we assumeP_hψ−P_h⁻ψ has the following expression inτ_j (Phψ−P_h⁻ψ)(x, t) =

∑k n=0

bnϕj,n(x, t), x∈τj. Recalling the definitions ofPhandP_h⁻ in (22) -(23), we have

bn =

∫

τ_jω(Phψ−P_h⁻ψ)ϕj,ndx

∫

τjωϕ_j,nϕ_j,ndx = 2 h_jγ_n

∫

τj

ω(ψ−P_h⁻ψ)ϕj,ndx

for alln≤k−1, and

b_k =−

k−1

∑

n=0

b_n ϕ¯_j,n(1) ϕ¯j,k(1).

Noticing that (ψ−P_h⁻ψ)⊥Pk−1, there holds for alln≤k−1,

|bn|= 2 hjγn

∫

τ_j

(ω−ω(xj))(ψ−P_h⁻ψ)ϕj,ndx

≲∥ω−ω(x_j)∥0,∞,τj∥ψ−P_h⁻ψ∥0,∞,τj∥ϕ_j,n∥0,∞,τj ≲h∥ω∥1,∞,τj∥ψ−P_h⁻ψ∥0,∞,τj, where in the last step we have used (19). Consequently,

|bk|≲

k−1

∑

n=0

|bn|≲h∥ω∥1,∞,τ_j∥ψ−P_h⁻ψ∥0,∞,τ_j,

and thus,

(28) ∥P_hψ−P_h⁻ψ∥0,∞,τj ≲

∑k n=0

|b_n|≲h∥ψ−P_h⁻ψ∥0,∞,τj.

Following the same line, we can prove

(29) ∥∂_t^m(Phψ−P_h⁻ψ)∥0,∞,τ_j ≲h∥∂_t^m(ψ−P_h⁻ψ)∥0,∞,τ_j, m≤k+ 1.

Then (24) follows from the standard approximation property of P_h⁻ (see, e.g., [21]).

Thanks to the super-closeness result (24), together with the superconvergence result forP_h⁻ (see, [7])

(ψ−P_h⁻ψ)(rj,i)≲h^k+2∥ψ∥k+2,∞, ∂x(ψ−P_h⁻ψ)(lj,i)≲h^k+1∥ψ∥k+2,∞,

9

the desired results (25)-(26) follow immediately.

Now we prove (27). On the one hand, using the orthogonality ofϕj,n, we get

∫

τ_j

ϕj,ndx=

∫

τ_j

ω⁻¹ωϕj,ndx=

∫

τ_j

(ω⁻¹−In−1ω⁻¹)ωϕj,ndx,

where I_nv ∈Pn is some interpolation function of v. Since ω =f^′(u) is sufficiently smooth,i.e.,|∂_xⁿω|≲1, we have

∥∂_xⁿω⁻¹∥0,∞,τ_j ≲ ∥ω∥n,∞,τ_j

∥ω∥ⁿ⁺¹0,∞,τj

≲1.

Then

∫

τ_j

ϕ_j,ndx

≲hⁿ⁺¹∥∂_xⁿω⁻¹∥0,∞,τj ≲hⁿ⁺¹.

On the other hand, by the orthogonality ofP_h⁻, that is, (ψ−P_h⁻ψ)⊥Pk−1, we have

|bn|=

∫

τj

ω(ψ−P_h⁻ψ)ϕj,ndx =

∫

τj

(ω−Ik−1−nω)(ψ−P_h⁻ψ)ϕj,ndx

≲h^2k+1−n∥ψ∥k+1,τ_j∥∂_x^k−nω∥0,τ_j ≲h^2k+1−n∥ψ∥k+1,τ_j. Noticing that

∫

τ_j

(ψ−Phψ)dx=

∫

τ_j

(P_h⁻ψ−Phψ)dx=−

∑k n=0

bn

∫

τ_j

ϕj,ndx,

then

∫

τ_j

(ψ−P_hψ)dx

≲h^2k+2∥ψ∥k+1,τj, which yields (27) directly. The proof is complete.

To end this subsection, we introduce a special operatorL, which will be used in later analysis. For allψ∈H_h¹, we define the operator L:H_h¹→Vhby

(f^′(u)L(ψ), v_x)_j = (ψ, v)_j, ∀v∈Pk(τ_j)\P0(τ_j), (30)

L(ψ)(x⁻_j+1 2

) = 0, ∀j ∈ZN. (31)

Noticing that vx ∈Pk−1(τj) for all v∈Pk(τj)\P0(τj), we have from the definition ofP_h

(f^′(u)P_hψ, v_x)_j = (ωP_hψ, v_x)_j = (ωψ, v_x)_j, v∈Pk(τ_j)\P0(τ_j).

Then the definition of L is similar to that of P_h except the term in the right-hand side (withωv_xreplaced byv). Therefore, we can similarly prove that the operatorL is well-defined.

3.2. Construction of u_I and its approximation properties. We first con- struct a series of functions starting withw₀=u−P_huand define

(32) wi+1=L(∂twi), 1≤i≤k.

We have the following properties.

10

Lemma 4. Let ω=f^′(u)≥δ >0be a sufficiently smooth function satisfying (33) |∂_x^mf^′(u)|≲1 |∂_t^mf^′(u)|≲1, m≤k+ 1.

In each elementτ_j, we express w_i’s in terms of orthogonal basis {ϕ_j,n} as

w_i|τj =

∑k n=0

c_i,nϕ_j,n.

Then the coefficientci,n satisfies

(34) |∂_t^mc_i,n|≲hmax(k+1+i,2k+1−n)∥∂^m+i_t u∥k+1,∞, n≤k, m= 0,1.

Consequently, there hold form= 0,1, (35)

∥∂_t^mw_i∥0,∞,τj ≲h^k+i+1∥∂_t^m+iu∥k+1,∞,

1 hj

∫

τ_j

∂_t^mw_idx

≲h^2k+1∥∂_t^i+mu∥k+1,∞. Proof. We only prove (34)-(35) for m = 0 since the similar argument can be applied to m = 1. We first prove (34) by induction. As v ∈ Pk \P0, we have v_x∈Pk−1. Then we choosev_x=ϕ_j,n, n= 0, . . . , k−1 in (30) to obtain

ci+1,n=

∫

τ_j∂twiD_x⁻¹ϕj,ndx

∫

τjωϕ_j,nϕ_j,ndx = 2 h_jγ_n

∫

τj

∂twiD⁻_x¹ϕj,ndx, n≤k−1, (36)

ci+1,k=−

k−1

∑

n=0

ci+1,n

ϕ¯j,n(1) ϕ¯j,k(1), (37)

whereγ_n is given by (15), andD_x⁻¹vis a function defined by

(38) D⁻_x¹v

τj =

∫ x x_j

−1 2

v(x)dx.

SinceD_x⁻¹ϕj,n∈Pn+1(τj), we have, from the orthogonality ofPh, c_1,n= 2

hjγn

∫

τ_j

ω⁻¹ω∂_t(u−P_hu)D⁻_x¹ϕ_j,ndx

= 2

hjγn

∫

τ_j

(ω⁻¹−I_k−2−nω⁻¹)

ω∂_t(u−P_hu)D_x⁻¹ϕ_j,ndx.

Here againI_mω∈Pmdenotes the interpolation function ofω. By (33) and standard approximation theory,

∥ω⁻¹−Ik−2−nω⁻¹∥0,∞,τ_j ≲h^k−1−n∥∂_x^k−1−nω⁻¹∥0,∞≲h^k−1−n, which yields, together with (19) and (24),

|c_1,n|≲h^k−n−1∥∂_t(u−P_hu)∥0,∞,τj∥D_x⁻¹ϕ_j,n∥0,∞,τj

≲h^k−n(∥∂t(u−P_h⁻u)∥0,∞+|∂t(Phu−P_h⁻u)∥0,∞)∥ϕj,n∥0,∞≲h^2k+1−n∥∂tu∥k+1,∞ 11