Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension

Superconvergence of the local discontinuous Galerkin method for linear fourth-order time-dependent problems in one space dimension

XIONGMENG∗

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, People’s Republic of China

∗Corresponding author: [email protected] CHI-WANGSHU

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA [email protected]

AND

BOYINGWU

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, People’s Republic of China

[email protected]

[Received on 2 April 2011; revised on 13 September 2011]

In this paper we investigate the superconvergence of local discontinuous Galerkin (LDG) methods for solving one-dimensional linear time-dependent fourth-order problems. We prove that the error between the LDG solution and a particular projection of the exact solution,e¯_u, achieves

k+³₂

th-order supercon- vergence when polynomials of degreek(k1) are used. Numerical experiments withP^kpolynomials, with 1k3, are displayed to demonstrate the theoretical results, which show that the errore¯uactually achieves(k+2)th-order superconvergence, indicating that the error bound fore¯_uobtained in this paper is suboptimal. Initial boundary value problems, nonlinear equations and solutions having singularities, are numerically investigated to verify that the conclusions hold true for very general cases.

Keywords: local discontinuous Galerkin method; superconvergence; fourth-order problems; error estimates.

1. Introduction

In this paper we are interested in the superconvergence of local discontinuous Galerkin (LDG) methods for a class of one-dimensional linear fourth-order problems formulated as

ut+αux +βux x+ux x x x =0, (1.1)

whereαandβ are arbitrary constants. Note that, forβ > 0, there is an antidiffusion termβux x in the equation, which is, however, dominated by the higher-order diffusion termux x x x. The general problem (1.1) includes the following linear time-dependent biharmonic equation

ut+ux x x x =0, (1.2)

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and the linearized Cahn–Hilliard equation

ut+ux x +ux x x x =0, (1.3)

as its special cases.

The discontinuous Galerkin (DG) method is a class of finite element methods using discontinuous piecewise polynomials as the solution and the test spaces. It was first introduced by Reed & Hill (1973) for solving a first-order steady-state linear conservation law and later developed by Cockburn et al.

(Cockburn & Shu, 1989, 1998b; Cockburn et al., 1989, 1990) for solving time-dependent nonlinear equations. Motivated by the successful numerical experiments of Bassi & Rebay (1997) for the com- pressible Navier–Stokes equations, the LDG methods were developed for solving nonlinear convection–

diffusion equations (Cockburn & Shu, 1998a) containing second-order spatial derivatives in which L² stability and a suboptimalL²error estimates were obtained for linear equations with smooth solutions.

Later, the LDG methods were generalized to solve various partial differential equations (PDEs) involv- ing higher-order derivatives. For Korteweg-de Vries-type equations containing third-order derivatives, an LDG method was developed in Yan & Shu (2002a), where a suboptimal error estimate was proved for the linear case, and more recently, an optimalL²error estimate was obtained in Xu & Shu (in press).

In Yan & Shu (2002b), Xu & Shu (2004); Xu & Shu (2006) and Xiaet al.(2007), LDG techniques were developed for solving other types of high-order PDEs including the time-dependent biharmonic equa- tions, the fully nonlinear K(n,n,n)equations, the Kuramoto–Sivashinsky-type equations, the Cahn–

Hilliard-type equations and so on. In Dong & Shu (2009), optimal error estimates for the LDG method applied to the linear biharmonic equation and linearized Cahn–Hilliard-type equations were obtained in one dimension and in multidimensions for Cartesian and triangular meshes. For more details of the DG and LDG methods we refer to the lecture notes, Cockburn (1999), and review papers, Cockburn & Shu (2001); Xu & Shu (2010).

Apart from the LDG methods mentioned above, there are also other finite element methods in the lit- erature for solving fourth-order time-dependent problems. For example, Elliott & Zheng (1986) applied a conforming finite element method to the Cahn–Hilliard equation and obtained optimal error estimates inL²andL^∞norms provided the approximate solution is bounded inL^∞and the polynomial degree k3. Feng & Prohl (2004) applied a mixed finite element method for solving Cahn–Hillard equations on quasiuniform triangular meshes and obtained an optimal error estimate under minimum regularity assumptions on the initial data and the domain.

Adjerid & Issaev (2005) and Adjerid & Klauser (2005) showed that the LDG solution is super- convergent at Radau points for solving convection- or diffusion-dominant time-dependent equations.

Based on Fourier analysis, Cheng & Shu (2008, 2009) proved superconvergence of the DG and LDG solutions towards a particular projection of the exact solution in the case of piecewise linear polyno- mials on uniform meshes for the linear conservation law and heat equation, respectively. The results were later improved, using a different technique, in Cheng & Shu (2010) for arbitrary nonuniform reg- ular meshes and schemes of any order. In this paper we follow the approach in Cheng & Shu (2010) to obtain the superconvergence property of the LDG method for a class of fourth-order problems. An important motivation for studying such superconvergence is to set a firm theoretical foundation for the excellent behaviour of DG and LDG methods for long-time simulations, which have been repeatedly observed by practitioners. Indeed, if superconvergence for the error between the DG or LDG solution and a particular projection of the exact solution of the order

k+³₂

, with linear growth in time, can be shown for polynomials of degreek, then the error between the numerical solution and the exact solution does not grow for a long timet =O ₁

√h

, wherehis the mesh size (Cheng & Shu, 2010). The gen- eralization from first- and second-order equations in Cheng & Shu (2010) to the fourth-order equation

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in this paper involves several technical difficulties, including the estimate of different combinations of the LDG solution and auxiliary variables that approximate derivatives of different orders, and the design and analysis of a special operator to guarantee the superconvergence property of the initial condition.

This paper is organized as follows. In Section 2 we define the LDG scheme for fourth-order time- dependent problems, state the main results and present the details of the proof of the superconvergence property. In Section 3 various numerical experiments, including linear equations, nonlinear equations, initial boundary value problems and solutions having singularities, are shown to demonstrate that the conclusions hold true for very general cases. Concluding remarks and comments on future work are given in Section 4. The proofs for some of the technical lemmas are collected in Appendix.

2. LDG scheme for fourth-order problems

We consider the following linear fourth-order equation

ut+αux+βux x +ux x x x =0 (2.1a)

with initial condition

u(x,0)=u0(x) (2.1b)

and periodic boundary conditions

u(0,t)=u(2π,t). (2.1c)

We would like to remark that the assumption of periodic boundary conditions is for simplicity only and not essential: see Cheng & Shu (2010) for discussion related to initial boundary value problems for conservation laws.

2.1 The LDG scheme

We assume the following mesh to cover the computational domain I = [0,2π], consisting of cells Ij =

x_j−1 2,x_j+1

2

, for 1 j N, where 0=x1

2 <x3

2 <· · ·<x_N+1

2 =2π.

The cell centre is denoted byxj = x_j−1

2+x_j+1 2

2. We also sethj =x_j+1 2−x_j−1

2 andh=maxjhj. We denote by(vh)⁻_j+1

2

and(vh)⁺_j+1 2

the values ofvh at the discontinuity pointx_j+¹

2 from the left cell, Ij, and from the right cell, Ij+1, respectively. The following piecewise polynomial space is chosen as the finite element space:

V_h^k= {v:v|I_j ∈ P^k(Ij),j =1, . . . ,N},

whereP^k(Ij)denotes the set of polynomials of degree up tokdefined on the cellIj. Note that functions inV_h^k are allowed to have discontinuities across element interfaces.

In order to construct the LDG scheme, firstly, we introduce some auxiliary variables approximating various order derivatives of the solution and rewrite equation (2.1a) as a first-order system,

ut+(αu+βq+r)x =0, r−px =0, p−qx =0, q−ux =0.

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Then, the semidiscrete LDG scheme is defined as follows: finduh,qh,ph,rh∈V_h^k, such that

Ij

(uh)tρdx−

Ij

αuhρxdx+αu˜hρ⁻|_j+¹

2 −αu˜hρ⁺|_j−¹

2 −

Ij

βqhρxdx

+βqˆhρ⁻|_j+¹

2 −βqˆhρ⁺|_j−¹

2 −

Ij

rhρxdx+ ˆrhρ⁻|_j+¹

2 − ˆrhρ⁺|_j−¹

2 =0, (2.2a)

I_j

rhηdx+

I_j

phηxdx− ˆphη⁻|_j+¹

2 + ˆphη⁺|_j−¹

2 =0, (2.2b)

Ij

phξdx+

Ij

qhξxdx− ˆqhξ⁻|_j+¹

2 + ˆqhξ⁺|_j−¹

2 =0, (2.2c)

Ij

qhψdx+

Ij

uhψxdx− ˆuhψ⁻|_j+¹

2 + ˆuhψ⁺|_j−¹

2 =0 (2.2d) hold for anyρ, ψ, ξ, η∈V_h^k,whereu˜his the upwind flux depending on the sign ofα. Without loss of generality we assume thatα0 and takeu˜h=u⁻_h and then choose alternating fluxes for the diffusion terms as follows:

ˆ

uh=u⁻_h, qˆh=q_h⁺, pˆh =p⁻_h, rˆh=r_h⁺. (2.3) 2.2 Notation and auxiliary results

To prove superconvergence of the LDG method, we would like to introduce the following notation, definitions and useful lemmas.

2.2.1 Notation for the DG discretization. First, we use [ξ] = ξ⁺−ξ⁻to denote the jump in the functionξat each cell boundary point. For the linear problems discussed in this paper, we introduce the DG discretization operatorDas in Xu & Shu (in press): for each cellIj =

x_j−1 2,x_j+1

2

,

DI_j(ξ, η; ˆξ)= −

I_jξηxdx+ ˆξη⁻|_j+¹

2 − ˆξη⁺|_j−¹

2. We also use the notation

D(ξ, η; ˆξ)=

j

DIj(ξ, η; ˆξ).

Using the definition of this operator, we have the following lemmas, whose proof is straightforward (see Xu & Shu, in press and also Zhang & Shu, 2010).

LEMMA2.1 (Xu & Shu, in press) Choosing different numerical fluxes the DG discretization operator satisfies the equalities

D(ξ, η;ξ⁻)+D(η, ξ;η⁺)=0, (2.4a)

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D(ξ, η;ξ⁺)+D(η, ξ;η⁻)=0, (2.4b) D(ξ, η;ξ⁺)+D(η, ξ;η⁺)= −

j

[ξ]_j+¹

2[η]_j+¹

2, (2.4c)

D(ξ, η;ξ⁻)+D(η, ξ;η⁻)=

j

[ξ]_j+1 2[η]_j+¹

2, (2.4d)

D(ξ, ξ;ξ⁻)=1 2

j

[ξ]²_j+1

2, (2.4e)

D(ξ, ξ;ξ⁺)= −1 2

j

[ξ]²

j+¹₂. (2.4f)

LEMMA2.2 By integration by parts we also have DI_j(ξ, η;ξ⁻)=

Ij

ξxηdx+[ξ]η⁺|_j−¹

2, (2.5)

DI_j(ξ, η;ξ⁺)=

I_jξxηdx+[ξ]η⁻|_j+¹

2. (2.6)

2.2.2 Projections and interpolation properties. In what follows we define two special projections, P_h^± into V_h^k, which are commonly used in the analysis of DG methods. For any given functionu ∈ H¹(I)and arbitrary subinterval Ij =

x_j−1 2,x_j+1

2

,the special projections ofu, denoted by P_h⁺uand P_h⁻u, are the unique functions in the finite element spaceV_h^ksatisfying, for each j,

Ij

(P_h⁺u(x)−u(x))ρ(x)dx=0 ∀ρ∈ P^k−1(Ij), (P_h⁺u)⁺_j−1 2

=u x_j−1

2

, (2.7)

Ij

(P_h⁻u(x)−u(x))ρ(x)dx=0 ∀ρ∈ P^k−1(Ij), (P_h⁻u)⁻_j+1 2

=u x_j+1

2

. (2.8)

For the special projections mentioned above, we have, by the standard approximation theory (Ciarlet, 1978), that

P_h^±u(·)−u(·)_L² C h^k+1, (2.9)

where both here and belowCis a positive constant (which may have a different value at each occurrence) depending solely onu and its derivatives but independent ofh.In particular, in equation (2.9),C = Cuk+1, whereuk+1is the standard Sobolev(k+1)norm andC is a positive constant independent ofu.

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In the proof of the error estimates the following inverse properties are needed: for anyvh∈V_h^kthere exists a positive constantCindependent ofh, such that

vh_Γ C h⁻¹²vhL², (2.10)

∂xvhL² C h⁻¹vhL², (2.11)

wherevh_Γ is the usualL²norm on the cell interfaces of the mesh.

2.2.3 Functionals related to the L²norm. To get the superconvergence property of the method two functionals related to theL²norm of a function onIj are needed as defined in Cheng & Shu (2010):

B⁻_j(f)=

Ij

f(x)x−x_j−1 2

hj

d

dx f(x)x−xj

hj

dx,

B⁺_j(f)=

I_j

f(x)x−x_j+1 2

hj

d

dx f(x)x−xj

hj

dx.

The functionals defined above have the following properties, which are essential to the proof of super- convergence.

LEMMA2.3 (Cheng & Shu, 2010) For any function f(x)∈C¹onIj we have B⁻_j (f)= 1

4hj

Ij

f²(x)dx+ f²(x_j+¹

2)

4 , (2.12)

B⁺_j (f)= − 1 4hj

Ij

f²(x)dx− f²(x_j−¹

2)

4 . (2.13)

The proof of this lemma is straightforward; see Cheng & Shu (2010).

2.2.4 Initial condition. To obtain the superconvergence property of the method, the initial condition of the numerical scheme should be chosen carefully to be compatible with the superconvergence error estimate. To this end we define an operator P_h^∗ as follows: for any functionu, then P_h^∗u ∈ V_h^k, and supposeqh,ph,rh∈V_h^kare the unique solutions (with given P_h^∗u) to

I_j

rhηdx+

I_j

phηxdx−p⁻_hη⁻|_j+¹

2 +p⁻_hη⁺|_j−¹

2 =0, (2.14a)

I_j

phξdx+

I_j

qhξxdx−q_h⁺ξ⁻|_j+¹

2 +q⁺_hξ⁺|_j−¹

2 =0, (2.14b)

Ij

qhψdx+

Ij

P_h^∗uψxdx−(P_h^∗u)⁻ψ⁻|_j+¹

2 +(P_h^∗u)⁻ψ⁺|_j−¹

2 =0, (2.14c)

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for anyψ, ξ, η∈V_h^k, then we require

I_j((P_h⁻u−P_h^∗u)−(P_h⁺q−qh)+(P_h⁺r−rh))ρdx=0 (2.15) for anyρ∈ P^k−1onIj and

(P_h⁻u−P_h^∗u)⁻=(P_h⁺q−qh)⁺−(P_h⁺r−rh)⁺ atx_j−¹

2. (2.16)

For the regular mesh considered in this paper we denoteλ=maxjhj/minjhj, which is a constant during mesh refinements. As to the operator defined above we have the following lemma.

LEMMA2.4 P_h^∗uexists and is unique. Moreover, there holds the error estimate

P_h⁻u−P_h^∗uL² C(λ,uk+4)h^k+3/2. (2.17) The proof of this lemma is given in Appendix.

We would like to remark that the purpose for introducing the operator P_h^∗is only theoretical: it is needed for the technical proof of superconvergence. In actual numerical computation we have observed that we can use the usualL²projection ofuas the initial condition and still observe superconvergence;

see the numerical experiments in Section 3. Of course, if the standard L² projection is used for the initial condition, then the superconvergence result does not hold att =0 nor for smallt. For later time the dissipativity in the PDE and the numerical scheme seems to help to recover the superconvergent performance.

2.3 Main results

Before we state the main results we would like to introduce the following notation:

eu=u−uh=(u−P_h⁻u)+(P_h⁻u−uh)=εu+ ¯eu, eq=q−qh=(q−P_h⁺q)+(P_h⁺q−qh)=εq+ ¯eq, ep= p−ph=(p−P_h⁻p)+(P_h⁻p−ph)=εp+ ¯ep,

er =r−rh=(r−P_h⁺r)+(P_h⁺r−rh)=εr + ¯er. For the caseα0 we have the following error estimates.

THEOREM2.5 Letu,p=ux xbe the exact solution of the fourth-order problem (2.1), which is assumed to be sufficiently smooth, i.e.uk+4,utk+4,uttk+4andutttk+1are bounded uniformly for any time t ∈ [0, T]. Let uh, ph be the LDG solution of equation (2.2) when the diffusion alternating fluxes (2.3) are used. We choose the initial condition asuh(·,0)= P_h^∗u0. For regular triangulations of I = [0,2π], if the finite element space V_h^k withk 1 is used, then there holds the following error estimate:

¯eu(·,t)²_L2 + _t

0 ¯ep(·,t)²_L2dt Ce^2C1th^2k+3, (2.18)

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and, in particular,

¯eu(·,t)_L² Ce^C1th^k+3/2,

whereC =C(α, β, λ,uk+4,utk+4,uttk+4,utttk+1)and both here and belowC1=C1(α, β)

0.

REMARK2.6 For the caseα0 we can chooseu˜h=u⁺_h and take the diffusion alternating fluxes as ˆ

uh=u⁺_h, qˆh=q_h⁻, pˆh =p⁺_h, rˆh=r_h⁻. (2.19) Theorem 2.5 still holds in this case with the obvious change of the projections.

For the caseα=β =0 equation (2.1a) reduces to the biharmonic equation (1.2), and we have the following result.

THEOREM2.7 Letu, p = ux x be the exact solution of the fourth-order problem (1.2), which is as- sumed to be sufficiently smooth, i.e.uk+4,utk+4,uttk+4andutttk+1are bounded uniformly for any timet ∈ [0, T]. Letuh, ph be the LDG solution of equation (2.2) whenα=β =0 and dif- fusion alternating fluxes (2.3) are used. We choose the initial condition asuh(·,0)=P_h^∗u0. For regular triangulations of I = [0,2π], if the finite element spaceV_h^k withk 1 is used, then there holds the following error estimate:

¯eu(·,t)²_L2+ _t

0 ¯ep(·,t)²_L2etC(1+t)²h^2k+3, and, in particular,

¯eu(·,t)L² C(1+t)h^k+3/2, whereC =C(λ,uk+4,utk+4,uttk+4,utttk+1).

The proof of this theorem is similar to that for the previous theorem, except that we need to carefully evaluate and estimate several terms to obtain a linear growth bound without employing Gronwall’s inequality. The detailed proof is given in Appendix.

The case with the fluxes (2.19) is the same with the obvious change of the projections.

Note that, for the general cases including the antidiffusive caseβ >0, the exponential growth of the constant with respect to time in Theorem 2.5 is expected, as the exact solution may have such growth in time for small wave numbers.

2.4 Proof of Theorem2.5

By using the DG discretization operator, the LDG scheme (2.2) with the fluxes (2.3) can be written as

I_j

(uh)tρdx+αDIj(uh, ρ;u⁻_h)+βDIj(qh, ρ;q_h⁺)+DIj(rh, ρ;r_h⁺)=0, (2.20a)

Ij

rhηdx−DIj(ph, η; p_h⁻)=0, (2.20b)

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I_j

phξdx−DI_j(qh, ξ;q_h⁺)=0, (2.20c)

I_j

qhψdx−DI_j(uh, ψ;u⁻_h)=0, (2.20d)

for anyρ, ψ, ξ, η∈V_h^k.Since the exact solutionsu,q=ux,p=ux x,r =ux x xalso satisfy the scheme (2.2), we have therefore the error equations

I_j(eu)tρdx+αDI_j(eu, ρ;e⁻_u)+βDI_j(eq, ρ;e_q⁺)+DI_j(er, ρ;e⁺_r )=0,

I_j

erηdx−DI_j(ep, η;e⁻_p)=0,

I_j

epξdx−DIj(eq, ξ;e⁺_q)=0,

Ij

eqψdx−DIj(eu, ψ;e⁻_u)=0,

which, by the properties of the projectionsP_h⁻andP_h⁺, given in equations (2.7) and (2.8), is

Ij

(eu)tρdx+αDIj(¯eu, ρ; ¯e⁻_u)+βDIj(¯eq, ρ; ¯e_q⁺)+DIj(¯er, ρ; ¯e⁺_r )=0, (2.21a)

Ij

erηdx−DI_j(¯ep, η; ¯e⁻_p)=0, (2.21b)

Ij

epξdx−DI_j(¯eq, ξ; ¯e⁺_q)=0, (2.21c)

I_j

eqψdx−DI_j(¯eu, ψ; ¯e⁻_u)=0, (2.21d)

for anyρ, ψ, ξ, η∈V_h^k.Taking(ρ, ψ, ξ, η)=(¯eu,−¯er,e¯p,e¯q)in equation (2.21), adding them up and summing over all j, we obtain

I

(¯eu)te¯udx+

I

¯ e²_pdx+

I

(εu)te¯udx+

I

εpe¯pdx+

I

εre¯qdx−

I

εqe¯rdx+αD(¯eu,e¯u; ¯e_u⁻)

+βD(¯eq,e¯u; ¯e_q⁺)+D(¯er,e¯u; ¯e⁺_r )+D(¯eu,e¯r; ¯e_u⁻)−D(¯eq,e¯p; ¯e⁺_q)−D(¯ep,e¯q; ¯e⁻_p)=0.

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Using the property of the operatorDin Lemma 2.1 we thus have

I

(e¯u)te¯udx+

I

¯ e²_pdx+

I

(εu)te¯udx+

I

εpe¯pdx

+

I

εre¯qdx−

I

εqe¯rdx+α 2

j

[e¯u]²

j+¹2 +βD(¯eq,e¯u; ¯e_q⁺)=0. (2.22) By takingξ = ¯euin equation (2.21c) and summing over all jwe get

D(¯eq,e¯u; ¯e_q⁺)=

I

epe¯udx. (2.23)

Combining equations (2.22) and (2.23) we arrive at 1

2 d

dt¯eu²_L2+¯ep²_L2 −

I(εu)te¯udx−

Iεpe¯pdx−

Iεre¯qdx+

Iεqe¯rdx−β

I

epe¯udx. (2.24) It follows from the Cauchy–Schwarz inequality that

1 2

d

dt¯eu²_L2 +1

2¯ep²_L2

I

(εu)te¯udx +

I

εpe¯pdx +

I

εre¯qdx +

I

εqe¯rdx + |β|

I

εpe¯udx +β²

2 ¯eu²_L2. (2.25) On the other hand, using Lemma 2.2, equation (2.21) can be rewritten as

I_j

(eu)tρdx+αDIj(¯eu, ρ; ¯e⁻_u)+

I_j

(e¯r +βe¯q)xρdx+[¯er +βe¯q]ρ⁻|_j+¹

2 =0, (2.26a)

Ij

erηdx−

Ij

(¯ep)xηdx−[e¯p]η⁺|_j−¹

2 =0, (2.26b)

Ij

epξdx−

Ij

(e¯q)xξdx−[e¯q]ξ⁻|_j+¹

2 =0, (2.26c)

Ij

eqψdx−

Ij

(¯eu)xψdx−[¯eu]ψ⁺|_j−¹

2 =0. (2.26d) Denote

¯

eu=rj +dj(x)(x−xj)/hj, e¯q=bj+sj(x)(x−xj)/hj,

¯

ep=vj+wj(x)(x−xj)/hj, e¯r =lj +gj(x)(x−xj)/hj,

where rj,bj, vj,lj are constants and dj(x),sj(x), wj(x),gj(x) ∈ P^k−1. First, taking ψ = dj(x) (x−x_j−1

2)/hj in equation (2.26d), and using the definition ofB⁻_j, we have

Ij

eqdj(x)(x−x_j−1

2)/hjdx−B⁻_j(dj)=0.

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By the property ofB⁻_j in Lemma 2.3 we obtain

Ij

d²_j(x)dx 4

Ij

eqdj(x)(x−x_j−1 2)dx.

Defining piecewise polynomialsd(x)andφ1(x), such thatd(x)=dj(x)andφ1(x)=x−x_j−1 2 onIj, and summing the above inequality over j, we get

d_L² 4eq_L²φ1L^∞ 4heq_L², (2.27) where we have used the fact thatφ1L^∞ =h.Similarly, takingξ =sj(x)(x−x_j+1

2)/hj in equation (2.26c),η=wj(x)(x−x_j−1

2)/hj in equation (2.26b) and using the definition ofB⁻_j andB⁺_j, we have

Ij

erwj(x)(x−x_j−¹

2)/hjdx−B⁻_j(wj)=0,

Ij

epsj(x)(x−x_j+1

2)/hjdx−B⁺_j (sj)=0.

By the properties ofB⁻_j andB⁺_j in Lemma 2.3 we obtain

I_j

w²_j(x)dx 4

I_j

erwj(x)(x−x_j−1 2)dx,

I_j

s²_j(x)dx −4

I_j

epsj(x)(x−x_j+¹

2)dx.

Defining piecewise polynomialsw(x),s(x)andφ2(x), such that w(x) = wj(x),s(x) = sj(x)and φ2(x)=x−x_j+1

2 onIj, and summing the above inequality over j, we get

w_L² 4er_L²φ1L^∞4her_L², (2.28) sL² 4epL²φ2L^∞4hepL², (2.29) where we have used the fact that φ1L^∞ = φ2L^∞ = h. Then, letting ρ = (gj(x)+βsj(x)) (x−x_j+1

2)/hj in equation (2.26a), we have

Ij

(eu)tρdx−α

Ij

¯

euρxdx+ ¯e_u⁻ρ⁺|_j−¹

2

+

Ij

(e¯r +βe¯q)xρdx =0,

which can be written as

Ij

(eu)t(gj(x)+βsj(x))(x−x_j+1

2)/hjdx−αR¹_j +R²_j =0,

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where

R¹_j =

I_je¯u((gj(x)+βsj(x))(x−x_j+1

2)/hj)xdx

−

rj−1+1 2dj−1

x_j−1

2

× gj

x_j−1

2

+βsj

x_j−1

2

=

Ij

dj(x)x−xj

hj ((gj(x)+βsj(x))(x−x_j+1

2)/hj)xdx +

rj−rj−1−1 2dj−1

x_j−1

2 gj

x_j−1

2

+βsj

x_j−1

2

and

R²_j =B⁺_j (gj+βsj)= − 1 4hj

Ij

(gj(x)+βsj(x))²dx−1 4

gj

x_j−1

2

+βsj

x_j−1

2

2

.

Therefore,

Ij

(gj(x)+βsj(x))²dx4

Ij

(eu)t(gj(x)+βsj(x))(x−x_j+1

2)dx−4αhjR¹_j. Summing the above inequality over all jwe arrive at

g+βs²_L2 4(eu)tL²g+βsL²φ2L^∞+4α

j

hjR¹_j

. (2.30)

Takingψ=1 in equation (2.26d) we get rj−rj−1−1

2dj−1

x_j−1

2

=

Ij

eqdx−1 2dj

x_j+1

2

.

So, the termhjR¹_j can be formulated as hjR¹_j =

I_j

dj(x)(x−xj)(gj(x)+βsj(x))/hjdx +

Ij

dj(x)(x−xj)(gj(x)+βsj(x))(x−x_j+¹

2)/hjdx +hj

Ij

eqdx−1 2dj

x_j+1

2 gj

x_j−1

2

+βsj

x_j−1

2

.

By the inverse properties (2.10) and (2.11) we have the estimate

j

hjR¹_j

C(k)g+βsL²(dL²+heqL²), (2.31)

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wherekis the degree of polynomials in the finite element spaceV_h^k. Combining equations (2.27), (2.30) and (2.31) together and recalling thatφ2L^∞ = x−x_j+1

2L^∞=h, we conclude that

g+βsL² C(α,k)h((eu)tL²+ eqL²). (2.32) Thus,

gL² g+βsL² + |β|sL² C(α, β,k)h((eu)tL²+ eqL²+ epL²). (2.33) Now, we return to the error equation (2.25). Note that(εu)t, εq, εpandεr are orthogonal to any piece- wise constant functions, then

I

(εu)te¯udx =

j

Ij

(εu)tdj(x)(x−xj)/hjdx

(εu)tL²dL²φL^∞,

I

εpe¯pdx =

j

Ij

εpwj(x)(x−xj)/hjdx

εpL²wL²φL∞,

Iεre¯qdx =

j

I_jεrsj(x)(x−xj)/hjdx

εrL²sL²φL^∞,

I

εqe¯rdx =

j

I_j

εqgj(x)(x−xj)/hjdx

εq_L²g_L²φL^∞,

I

εpe¯udx =

j

Ij

εpdj(x)(x−xj)/hjdx

εpL²dL²φL^∞, whereφ=(x−xj)/hj. Then, equation (2.25) becomes

1 2

d

dt¯eu²_L2 +1

2¯ep²_L2 φL^∞

(εu)tL²dL²+ εpL²wL²+ εrL²sL²

+ εqL²gL²+ |β|εpL²dL²

+β² 2 ¯eu²_L2. Using the approximation property of the projections (2.9) and the fact thatφL^∞ = ¹₂, we get

1 2

d

dt¯eu²_L2 +1

2¯ep²_L2 C h^k+1(dL² + sL²+ wL²+ gL²)+β²

2 ¯eu²_L2, (2.34) where C = C(β,uk+4,utk+1). Substituting equations (2.27), (2.28), (2.29) and (2.33) into equation (2.34), we obtain

1 2

d

dt¯eu²_L2 +1

2¯ep²_L2 C h^k+2((eu)t_L²+ eq_L²+ ep_L² + er_L²)+β²

2 ¯eu²_L2, (2.35)

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