Local Analysis of Local Discontinuous Galerkin Method for the Time-Dependent Singularly Perturbed Problem

DOI 10.1007/s10915-014-9901-6

Local Analysis of Local Discontinuous Galerkin Method for the Time-Dependent Singularly Perturbed Problem

Yao Cheng · Feng Zhang· Qiang Zhang

Received: 11 May 2014 / Revised: 25 July 2014 / Accepted: 2 August 2014 / Published online: 17 August 2014

© Springer Science+Business Media New York 2014

Abstract In this paper we will present the local stability analysis and local error estimate for the local discontinuous Galerkin (LDG) method, when solving the time-dependent singularly perturbed problems in one dimensional space with a stationary outflow boundary layer. Based on a general framework on the local stability, we obtain the optimal error estimate out of the local subdomain, which is nearby the outflow boundary point and has the width of O(h log(1/h)), for the semi-discrete LDG scheme and the fully-discrete LDG scheme with the second order explicit Runge–Kutta time-marching. Here h is the maximum mesh length.

The numerical experiments are given also.

Keywords Local stability analysis·Local error estimate·Local discontinuous Galerkin method·Singularly perturbed·Outflow boundary layer

Mathematics Subject Classification 65M12·65M15·65M60

1 Introduction

In this paper we are interested in the numerical performance of local discontinuous Galerkin (LDG) method to solve the time-dependent singularly perturbed problem in one dimensional space

u_t−εu_{x x}+u_x+c(x,t)u= f(x,t), (x,t)∈(0,1)×(0,T], (1.1)

Research supported by NSFC grant 11271187.

Y. Cheng·F. Zhang·Q. Zhang (

B

Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China e-mail: qzh@nju.edu.cn

Y. Cheng

e-mail: ycheng@smail.nju.edu.cn F. Zhang

e-mail: zhangfeng@smail.nju.edu.cn

whereε >0 is a very small diffusion parameter. Here c(x,t)and f(x,t)are smooth and bounded functions, and T >0 is the final time. When equipped with the Dirichlet boundary condition, the solution of (1.1) often varies quickly with a huge gradient and forms an outflow boundary layer near the outflow boundary point x=1.

This serious phenomenon often causes the numerical difficulties, so many numerical meth- ods on this problem have been presented and developed [11]. For example, limited in finite element methods, there are the standard finite element method combined with special mesh [15,24], streamline upwinding Petrov-Galerkin method [7,9,14], interior penalty discontin- uous Galerkin method [6,20] and LDG method [2,16,25]. The LDG method was introduced by Cockburn and Shu in [3] as an extension of Runge–Kutta discontinuous Galerkin meth- ods for conservation laws to the general convection-diffusion problems, even to those partial differential equations with high-order derivatives. For a fairly complete set of references on the discontinuous Galerkin methods as well as their implementation and applications, please see the review papers [5,13] and the recent book [8].

The LDG method is very good at solving numerically those fast-varying solutions, even those discontinuous solutions. This advantage has been showed by many numerical exper- iments. However, from the theoretical viewpoint, there are not enough support in stability and error estimate. Till now, many conclusions are restricted in the global analysis, namely, the solution is assumed to be smooth enough in the whole domain. For example, the semi- discrete LDG method for convection diffusion problems with periodic boundary conditions was considered in [3], and the quasi-optimal accurate was obtained. After that, Castillo [2]

developed the error estimate to the optimal order, in the hp-version of the LDG method for convection-diffusion problems with Dirichlet boundary condition. Recently, Wang and Zhang [16] considered the fully-discrete LDG scheme, where the authors employed the third order explicit Runge–Kutta time-marching and presented the correction on the reduction of accuracy due to the boundary setting on each stage time. The other work related to this topic include the super-convergence study of discontinuous Galerkin method on the special meshes, for example, in [17,18].

However, for the singularly perturbed problems, the exact solution in general does not have the global and uniform smoothness in the whole domain. As a consequence, the global results mentioned above look a little useless when the diffusion parameterεgoes to zero. Hence, we have to carry out the local analysis in the stability and error estimate, to describe clearly the numerical performance in a sub-domain of the LDG method. As far as the authors know, there are only couples of works on this issue. Guzmán [6] considered the interior penalty discontinuous Galerkin method for two dimensional steady problems, and obtained the quasi- optimal L²-norm local error estimate in the region that is away from the outflow boundary.

The width of cut-off region isO(h log(1/h))in the upwind direction, andO(h^1/2log(1/h)) in the crosswind direction, respectively. Here h is the maximum size of all elements. Recently, Zhu and Zhang in [25] considered really the LDG method for the one-dimensional steady problem, and presented the optimal local error estimate with the width of cut-off region in the orderO(h log(1/h)). In this paper, we are going to extend the above local analysis from the steady problem to the time-dependent problem. In this case, the smoothness of the exact solution of (1.1) becomes more complex.

As the first step of our study, we would like to consider the local stability analysis and local error estimate of LDG method for (1.1) with the stationary boundary layer in this paper. The moving interior layer will be considered in the further work. Associated with the time variable, we will consider in this paper both the semi-discrete LDG scheme and one version of fully- discrete LDG scheme, where the second order explicit Runge–Kutta (RK2) time-marching is coupled. The main technique used in this paper is the energy estimate with a suitable weighted

norm as in [6,23,25]. Compared with the analysis of the steady problem, the important point now is how to cope carefully with those terms involving the derivative (or finite differences) in time and the weight function, especially in the fully-discrete analysis; seeing Lemma4.1.

The highlight in this process is a full use of the highest-frequency component of piecewise polynomials, motivated by the work in [19,23]. This technique can also help us to obtain the local stability and local error estimate for the fully-discrete LDG scheme under the standard temporal-spatial condition. Furthermore, the Gauss–Radau projection also plays an important role to get the optimal local error estimate in this paper.

The rest of this paper is organized as follows. In Sect.2, we present the semi-discrete LDG scheme and one version of the fully-discrete scheme. In this paper we mainly consider the LDG method with piecewise linear polynomials and the RK2 time-marching, which is called as the LDGRK2 scheme. In Sect.3we present the weight function and some elemental properties of the finite element space and the LDG space discretization. The next two sections are the main body of this paper, where the local stability analysis and the local error estimate are given. In Sect.6, some numerical experiments are presented. Finally, we will end in Sect.7with concluding remarks.

2 LDG Schemes

Now we follow [2,16] and present the precise definition of the LDG method for the singularly perturbed convection-diffusion Eq. (1.1), with the initial solution u0(x)and the Dirichlet boundary condition

u(0,t)=a(t), u(1,t)=b(t), t∈(0,T]. (2.1) For convenience of analysis, in this paper we would like to consider the homogeneous bound- ary condition, namely, a(t)=b(t)=0 for any time t∈ [0,T].

Instead of directly discretizing (1.1), we would like to seek the numerical solution of the LDG method in the discontinuous finite element space through the equivalent first-order system

ut+(hu)x+cu= f, q+(hq)x =0, (2.2) where q = √

εux is the auxiliary variable. Here(hu,hq) = (u−√ εq,−√

εu) is the given flux.

2.1 Discontinuous Finite Element Space

LetTh = {Ij =(x_j−1/2,x_j+1/2)}^J_j=1be the partition of I =(0,1), where x_1/2 =0 and x_J+1/2 =1 are the real boundary points. Denote the cell length h_j =x_j+1/2−x_j−1/2for j=1, . . . ,J , and define h=max_jh_j. We assume_This quasi-uniform in this paper, namely, there exists a fixed positive constant C independent of h, such that C h≤hj ≤h for any j , as h goes to zero.

Associated with this mesh, we define the discontinuous finite element space

Vh = {p∈L²(I): p|Ij ∈Pk(Ij),j =1, . . .J}, (2.3) wherePk(I_j)denotes the space of polynomials in I_jof degree at most k≥1. Note that the functions in this space are allowed to have discontinuties across the element interfaces. As usual, we denote the jump at each interior interface point by

[[p]]_j+1/2=p⁺_j+1/2−p⁻_j+1/2, j=1,2, . . . ,J−1, (2.4)

where p⁺ and p⁻, respectively, are the traces from the right and the left direction. For notational convenience, we also denote[[p]]1/2 = p⁺_1/2 and[[p]]J+1/2 = −p⁻_J+1/2 at two actual boundary points.

For any piecewise H¹-function p, two kinds of Gauss–Radau projections [2], denoted by Php andNhp, respectively, will be used in this paper. Both of them are defined element by element as the unique function in V_h, such that

(Php)⁺_j−1 2

= p⁺

j−¹₂, Php−p,sh_Ij =0, ∀sh ∈Pk−1(Ij), ∀j; (2.5a) (Nhp)⁻_j+1

2

= p⁻

j+¹₂, Nhp−p,s_hI_j =0, ∀s_h ∈Pk−1(I_j), ∀j, (2.5b) hold respectively. Here·,·Ij is the standard inner-product in L²(I_j). Note that the exact collocation at one endpoint of each element helps us to get the optimal error estimate.

2.2 Semi-Discrete LDG Scheme

The semi-discrete LDG scheme is defined as follows. For any time t >0, we will seek the numerical solution wh=(uh,qh) ∈Vh×Vh, such that in each element Ij,j=1,2, . . . ,J , there hold

u_h,t,sh

I_j+ cuh,shIj − hu,s_h,x

I_j +(hus_h⁻)_j+¹

2 −(hus_h⁺)_j−¹

2 = f,shI_j, (2.6a) qh,rh_Ij −

hq,rh,x

Ij +(hqr_h⁻)_j+¹

2 −(hqr_h⁺)_j−¹

2 =0, (2.6b)

for any test function v_h =(s_h,r_h)∈V_h×V_h. Here the initial solution is taken as u_h(0)= Nhu0(x).

Noticing the expression of convection term in (1.1), we would like to follow [2,16] and define the numerical fluxes, bothhuandhq, in the form

h_u hq

j+¹₂ =

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

a(t)−√ εq_h⁺_,1

2

,−√ εa(t)

, j=0;

u⁻

h,j+¹₂ −√ εq⁺

h,j+¹₂,−√ εu⁻

h,j+¹₂

, 1≤ j ≤J−1;

(1+γ )u⁻_h,J+1 2

−γb(t)−√ εq_h,J+⁻ 1

2

,−√ εb(t)

, j=J;

(2.7) with the parameter

γ = ε

h, (2.8)

which is the inverse of the mesh Peclét number. This definition of numerical fluxes are very important to ensure the good stability and the optimal accuracy.

Till now we complete the definition of the semi-discrete LDG scheme for (1.1). However, for convenience of analysis, we would like to write the semi-discrete LDG scheme into a compact form. To that end, by·,·we denote

p,r = J

j=1

p,rI_j, p,r⁰

h =

J−1 j=1

p_j+1 2r_j+1

2, (2.9)

where⁰_h = {xj+1/2}^J−1_j=1 is the set of all interior element interface points. By summing up all the variation forms in (2.6) over all elements, and noticing the homogeneous boundary condition under our consideration, we arrive at the compact form of the above semi-discrete LDG scheme

u_h,t,s_h

+B_h(w_h,v_h)= f,s_h, ∀v_h =(s_h,r_h)∈V_h×V_h, (2.10) where the bilinear functional is defined as

B_h(w_h,v_h)= q_h,r_h + cu_h,s_h +B₁(u_h,q_h;s_h)+B₂(u_h,r_h), (2.11) and

B₁(u_h,q_h;s_h)= − u_h−√

εq_h,s_h,x

− u⁻_h −√

εq_h⁺,[[s_h]]

⁰_h

+(1+γ )u⁻

h,J+¹₂s⁻

h,J+¹₂ −√ εq⁻

h,J+¹₂s⁻

h,J+¹₂ +√ εq⁺

h,¹₂s⁺

h,¹₂, (2.12a) B2(uh,rh)= √

εuh,rh,x

+√ ε

u⁻_h,[[rh]]

⁰_h. (2.12b)

2.3 Fully-Discrete LDG Scheme

In this paper we also consider one fully-discrete LDG scheme, which adopts the second order total variation diminishing explicit Runge–Kutta time-marching; seeing [1,13].

To do that, let{tⁿ =nτ}^N_n=0be a uniform partition of the time interval[0,T], with the time stepτ = T/N . Remark that the time step could actually change step by step, but we take the time step as a constant for simplicity in this paper.

We can update the solution as follows. Assume wⁿ_h = (uⁿ_h,q_hⁿ) has been obtained at the current time tⁿ, we can find the numerical solution at the next time tⁿ⁺¹, through the intermediate solution w^n,1_h = (u^n,1_h ,q_h^n,1). They all belong to the finite element space V_h×V_h, and satisfy

u^n,1_h ,s_h

= uⁿ_h,s_h

−τB_h(wⁿ_h,v_h)+τ fⁿ,s_h

; (2.13a)

uⁿ⁺¹_h ,sh

= 1 2

uⁿ_h+uⁿ_h^,1,sh

−τ

2Bh(wⁿ_h^,1,vh)+τ 2

fⁿ⁺¹,sh

, (2.13b)

for any test function vh = (sh,rh) ∈ Vh ×Vh. The initial solution is taken the same as before, namely u⁰_h =Nhu0.

Considering the practical status, we would like in this paper to focus on the piecewise linear polynomials. The corresponding scheme (2.13) is referred to as the LDGRK2 scheme.

To ensure the numerical stability, the time step should satisfy a suitable temporal-spatial conditionτ ≤ λh, whereλ is a given CFL number. In this paper we do not pay more attention on the sharpest setting ofλ.

Remark 2.1 If we use the higher-order piecewise polynomials in scheme (2.13), a stronger CFL condition, for example τ = O(h^4/3), is necessary to ensure the stability, since the considered problem is convection-dominated. Please refer to [21] for the discussion about the pure hyperbolic equations.

3 Preliminaries

In the rest part of this paper, we will devote us to establishing the local stability and local error estimate of the above two schemes. In this section we present some elemental properties that will be used many times.

3.1 Weight Function

In this paper we follow [23] and introduce the weight function in the form ψ(x)=

e^{−(x−xc)/(K h)}, x>xc,

2−e^{(x−xc)/(K h)}, x≤x_c,, (3.1) where K ≥1 describes the steepness of the weight function, and x_cis the steepness center.

Here, xcis taken as a fixed constant independent of the time t, since the outflow boundary layer does not move.

It is easy to see that the weight function (3.1) is bounded and non-increasing. Furthermore, this weight function satisfies

|ψ(x)| ∈

[1,2), x ≤xc,

(0,h^α], x ≥xc+αK h log¹_h, (3.2) for arbitrary numberα≥0, and

|ymax|≤K h

D^mψ(x+y) D^mψ(x)

≤e, D^m+1ψ(x)≤K⁻¹h⁻¹D^mψ(x), (3.3) for m=0,1 almost everywhere. Here, D^mψdenotes the derivative ofψwith order m>0;

noting that D⁰ψ = ψ. The verification is straightforward and so omitted here; similar discussions can be found in [23].

Thanks to property (3.3), we can obtain the following inverse properties and approximation properties, measured in the weighted norm

ψp ≡

⎡

⎣^J

j=1

Ij

ψ²p²dx

⎤

⎦

1 2

, ψp_h ≡

⎧⎨

⎩ J

j=1

(ψp⁻)²_j+1

2 +(ψp⁺)²_j−1 2

⎫

⎬

⎭

1 2

, ∀p.

(3.4) For the notational convenience, below by the notation C we will denote a generic positive constant independent of h, τ,K andε⁻¹. It may have different value at each occurrence.

Lemma 3.1 (inverse properties) Let m be a positive integer. There exists an inverse constant C>0, such that

˜ψD^mp_h ≤C h^−m ˜ψp_h, ˜ψp_hh ≤C h^−1/2 ˜ψp_h, ∀p_h∈V_h, (3.5) whereψ˜ is one of the weight functionsψandψ⁻¹.

Lemma 3.2 (approximation property) For any smooth function p, there exists a constant C>0, such that

ψW^⊥_hp +hψD(W^⊥_h p) +h¹²ψW^⊥_hph ≤C h^min(k+1,m)ψD^mp, (3.6) where m is a given integer, and_W^⊥_h p= p−Whp is the projection error. Here_Wh is the Gauss–Radau projection, eitherPhorNh.

Lemma 3.3 (super-approximation properties) For any function ph ∈ Vh, there exists a constant C>0, such that

ψ⁻¹W^⊥_h(ψ²p_h) +h¹²ψ⁻¹W^⊥_h(ψ²p_h)_h ≤C h¹²K⁻¹²|ψψx|¹²p_h, (3.7a)

where K is the parameter in the weight function. As an application of weight property, we also have a rough estimate

ψ⁻¹W^⊥_h(ψ²p_h) +h¹²ψ⁻¹W^⊥_h(ψ²p_h)_h ≤C K⁻¹ψp_h, (3.7b) which implies the boundedness of the local projection, namely

ψ⁻¹Wh(ψ²p_h) ≤Cψp_h. (3.8)

HereWhis the Gauss–Radau projection, eitherPhorNh.

Similar discussions of the above conclusions can be also found in [23], so we omit the detailed proofs here. Note that the analysis in this paper depends strongly on the super- approximation property (3.7a), which is a little stronger than that in [23].

Remark 3.1 It is worthy to point out that the conclusions from Lemmas3.1to3.3also hold in any single element, and/or for the trivial weight function that is equal to one everywhere.

3.2 Properties of LDG Spatial Discretization

Now we present two important properties with respect to the bilinear functional B_h(·,·), when the weight function (3.1) is used and the diffusion parameter satisfiesε < h. To do that, we would like to introduce the main norm

v ≡

⎡

⎣|ψψx|²¹s²+ ψr²+1 2

0≤j≤J

[[ψs]]²_j+1

2 +γ[[ψs]]²_J+1 2

⎤

⎦

1 2

, (3.9)

for any piecewise H¹-function v=(s,r).

Lemma 3.4 Supposeε <h. For any piecewise H¹-function v=(s,r), there holds Bh(v, ψ²v)≥(1−K⁻¹²)v²−Cψs², (3.10) where the bounding constant C>0 is independent of K,h,v andε⁻¹.

Proof It is followed from (2.11) that Bh(v, ψ²v)= ψcs, ψs +1+2, where 1 = −

s, (ψ²s)x

−

s⁻,[[ψ²s]]

⁰_h+(1+γ )[[ψs]]²_J+1

2 + ψr², (3.11a) 2 =√

ε

r, (ψ²s)x

+√ ε

s, (ψ²r)x

+√ ε

r⁺,[[ψ²s]]

⁰_h+√ ε

s⁻,[[ψ²r]]

⁰_h

−√ εr⁻

J+¹₂(ψ²s)⁻_J+1

2 +√

εr⁺₁

2(ψ²s)⁺1

2. (3.11b)

Below we will analyze them one by one.

It is easy to see that| ψcs, ψs | ≤ Cψs², since the given function c is bounded.

Noticing the definition of jump on the boundary points, and the weight functionψis smooth everywhere, an integration by parts yields

−

s, (ψ²s)x

= − s, ψψxs +1 2

0≤j≤J

[[ψ²s²]]_j+¹

2

= − s, ψψxs +1 2

1≤j≤J−1

(s⁺

j+¹₂ +s⁻

j+¹₂)[[ψ²s]]_j+¹

2 +1

2[[ψs]]²1

2 −1

2[[ψs]]²_J+1 2.

Sinceψx < 0, the first term on the right is equal to|ψψx|^1/2s², and thus provides an additional numerical stability, which is stronger thanψs². After some manipulations on the jumps, we yield

1= v². (3.12a)

Along the same line, we also use the simple identity that[[pq]] = p⁻[[q]] + [[p]]q⁺at each interior element interface point, and get that

2=2√

εψxs, ψr.

Hence, we use the weighted Cauchy–Schwarz inequality, and the second inequality in prop- erty (3.3), to obtain

2≤ 2√

εK⁻¹²h⁻¹²|ψψx|¹²sψr

≤ √

εK⁻¹²h⁻¹²

|ψψx|¹²s²+ ψr²

≤K⁻¹²v², (3.12b) owing toε < h. This is the important treatment in the analysis to obtain the stability and error estimate in the optimal width of cutting-off region.

Finally, collecting up the above conclusions completes the proof of this lemma.

Lemma 3.5 Supposeε < h. For any function vh = (sh,rh) ∈ Vh ×Vh, there exists a bounding constant C>0 such that

Bh(vh,R^⊥_h(ψ²vh))≤C K⁻¹²

vh²+ ψsh²

. (3.13)

whereRh(ψ²vh)=(Nh(ψ²sh),Ph(ψ²rh))involves different Gauss–Radau projections.

Proof For notational convenience, we denote

R^⊥_h(ψ²v_h)=(N^⊥_h(ψ²s_h),P^⊥_h(ψ²r_h))=(s,˜ r)˜ , and split the considered term into the following form

Bh(vh,R^⊥_h(ψ²vh))=1+2+3+4, (3.14) where

1 = cs_h,s˜ + r_h,r˜ , (3.15a)

2 = √

εrh,s˜x +√ ε

r_h⁺,[[˜s]]

⁰_h−√ εr_h,J+⁻ 1

2

˜ s⁻

J+¹₂ +√ εr_h,⁺1

2

˜ s⁺₁

2

, (3.15b) 3 = √

εs_h,r˜_x +√ ε

s_h⁻,[[˜r]]

⁰_h, (3.15c)

4 = − s_h,s˜_x − s_h⁻,[[˜s]]

⁰_h+(1+γ )s⁻

h,J+¹₂s˜⁻

J+¹₂. (3.15d)

Each term on the right-hand side will be estimated below.

Firstly, by using the weighted Cauchy–Schwarz inequality, and the super-approximation property (3.7b) in Lemma3.3, we have

1 ≤ Cψshψ⁻¹s +˜ Cψrhψ⁻¹r ≤˜ C K⁻¹

ψsh²+ ψrh²

≤ C K⁻¹

ψs_h²+ v_h²

. (3.16a)

Secondly, noticing the definition of Gauss–Radau projections, we can eliminate those integrations in each element by an application of integration by parts. Hence

r_h,s˜_x = − r_h,x,s˜

− J

j=0

[[r_hs˜]]_j+¹

2 = −

J j=0

[[r_hs˜]]_j+¹

2

= − r_h⁺,[[˜s]]

⁰_h−

[[r_h]],s˜⁻

⁰_h+r⁻

h,J+¹₂s˜⁻

J+₂¹−r⁺

h,¹₂s˜⁺₁

2. Sinces˜⁻=0 at each element point, we have

2= −√ ε

[[rh]],s˜⁻

_h⁰=0. (3.16b)

Thirdly, along the similar line we have 3= −√

ε J

j=0

[[s_hr˜]]_j+¹

2 +√

ε s_h⁻,[[˜r]]

⁰_h =√

ε(s_h⁻)_J+¹

2r˜⁻

J+¹₂,

by noticingr˜⁺ =0 at each element interface point. Thus, we use the inverse property and the super-approximation property (3.7b) in Lemma3.3, to get that

3≤ C√

εh⁻¹²|[[ψsh]]|_J+¹

2ψ⁻¹r ≤˜ C√

εh⁻¹²K⁻¹|[[ψsh]]|_J+¹

2ψrh

≤ C√ εh⁻¹²

1

2+γ₋¹

2K⁻¹ 1

2+γ

|[[ψs_h]]|²_J+1

2 + ψr_h²

≤ C K⁻¹vh², (3.16c)

where we have used the simple fact that√

εh⁻¹²(1/2+γ )⁻¹² ≤√

2, sinceε <h.

Similarly as above, after some manipulations we also have that

4=

0≤j≤J−1

[[sh]]_j+¹

2s˜⁺

j+¹₂.

Then we use the weighted Cauchy–Schwarz inequality and the super-approximation property (3.7a) in Lemma3.3, to obtain

4 ≤

⎡

⎣

0≤j≤J−1

[[ψsh]]²_j+1 2

⎤

⎦

1

2 ⎡

⎣

0≤j≤J−1

(ψ⁻¹s˜⁺)²_j+1 2

⎤

⎦

1 2

≤ C K⁻¹²vh |ψψx|¹²sh ≤C K⁻¹²vh². (3.16d) This inequality is very important to obtain the sharp estimate to the width of narrow region nearby the outflow boundary points.

Finally, by collecting up the above estimates and noticing K⁻¹≤K⁻¹², we can complete

the proof of this lemma.

4 Local Stability Analysis

In this section we would like to obtain the local stability of the above two LDG schemes. To this end, we consider firstly the stability in the weighted L²-norm.

4.1 Semi-Discrete Scheme

For the generality, let us start from the following variation form similar as the semi-discrete scheme: find z_h =(βh, σh)∈V_h×V_hfor any time t∈(0,T], such that

β_h,t,s_h

+B_h(zh,v_h)=F_h(vh), ∀vh =(sh,rh)∈V_h×V_h, (4.1) where the initial solution is taken asβh(0)∈Vh.

Here B_h(·,·)is the bilinear functional defined in (2.11), and F_h(·)is a linear functional defined in the form

Fh(vh)=F1(sh)+F2(rh), ∀vh ∈Vh×Vh. (4.2) In what follows we assume the linear functional is bounded, namely, there are two given bounding constants Fand F, such that

|Fh(vh)| ≤Fvh, and |F1(sh)| ≤Fψ⁻¹sh, (4.3) where

v_h=

ψ⁻¹s_h²+ ψ⁻¹r_h²+1 2+γ

[[ψ⁻¹s_h]]²_J+1 2

¹

2. (4.4)

Note that the functional Fh should be chosen in different forms when we consider the stability and error estimate. The detailed representations will be given in the relative analysis;

please see the statement before Theorem4.1for stability analysis, and definition (5.6) for error estimate, respectively.

In order to analyze the local stability of problem (4.1), we would like to introduce the highest-frequency component of piecewise polynomials, inspired by [19,23]. For any g_h ∈ Vh, its highest-frequency component, denoted byM(gh)∈Vh, is defined element by element and satisfies in each element

M(gh),shI_j =0, ∀sh ∈Pk−1(Ij),∀j. (4.5) This is implemented easily by using the Legendre polynomials of degree up to k, which are orthogonal to each other in the standard inner-product of L²-space.

The following main lemma is a key-point in our analysis, especially, in the analysis for the fully-discrete LDG scheme.

Lemma 4.1 Assume zh=(βh, σh)∈Vh×Vhis given. Let gh∈Vhsatisfy

gh,sh +Bh(zh,vh)=Fh(vh), ∀vh =(sh,rh)∈Vh×Vh, (4.6) where Bh(·,·)and Fh(·)are the same as (4.1). If the parameter in the weight function, K≥1, is large enough, there holds

ψM(g_h)²≤Cψβh²+C h⁻¹z_h²+C F². (4.7) where the bounding constant C>0 is independent of h,K andε⁻¹.

Proof Using the orthogonal property (4.5), we can get easily that ψM(g_h)² =

M(g_h),N^⊥_h(ψ²M(g_h)) +

M(g_h),Nh(ψ²M(g_h))

=

M(gh),N^⊥_h(ψ²M(gh))

+ M(gh),th

=

M(g_h),N^⊥_h(ψ²M(g_h))

+ g_h,t_h,

(4.8)

where p_h=Nh(ψ²M(g_h))and t_h=M(p_h). Below we estimate two terms on the right-hand side of (4.8).

Before that, let us recall again the orthogonal property (4.5) in each element of highest- frequency component. Then a simple application of Cauchy–Schwarz inequality yields

M(ph)²_L2(Ij)= ph,M(ph)_Ij ≤ ph_L²_(Ij)M(ph)_L²_(Ij), ∀j,

which impliesM(ph)_L²_(Ij)≤ ph_L²_(Ij)for any j . Hence, there exists a bounding con- stant C>0 such thatψ⁻¹M(ph) ≤Cψ⁻¹ph, owing to the first inequality in property (3.3) with the weight functionψ⁻¹. This implies

ψ⁻¹th ≤Cψ⁻¹ph =Cψ⁻¹Nh(ψ²M(gh)) ≤CψM(gh), (4.9) due to (3.8). This inequality will be used several times below.

The first term is easy to estimate. As a direction application of weighted Cauchy–Schwarz inequality and the super-approximation property (3.7b) in Lemma3.3, we have

M(g_h),N^⊥_h(ψ²M(g_h))

≤C K⁻¹ψM(g_h)²≤ 1

8ψM(g_h)², (4.10) if K is large enough. However, the second term is a little complex to estimate. To do that, we have to employ (4.6) with a special test function, and get the splitting

g_h,t_h = − cβh,t_h −B₁(z_h,t_h)+F₁(t_h)≡Y₁+Y₂+Y₃,

in which each term above can be estimated by applications of the weighted Cauchy–Schwarz inequality and Young’s inequality. Noticing (4.9), we can get

Y₁≤Cψβhψ⁻¹t_h ≤CψβhψM(g_h) ≤ 1

8ψM(g_h)²+Cψβh². (4.11) As for the next term Y₂, we can make an integration by parts and achieve the splitting Y₂=Y₂₁+Y₂₂+Y₂₃, where

Y21= − (βh−√

εσh)x,th

=0, (4.12a)

due to the orthogonality property (4.5), since(βh−√

εσh)x is piecewise polynomials of degree of k−1. Also, we have

Y22= − J

j=0

[[βhth]]_j+¹

2 +

β_h⁻,[[th]]

_h⁰−(1+γ )(β_h⁻t_h⁻)_J+¹

2

= −

[[βh]],t_h⁺

_h⁰−γ (βhth)⁻_J+1 2

−(βhth)⁺1 2

,

after some simple manipulations. The weighted Cauchy–Schwarz inequality, together with the second inverse inequality in (3.5) with the weight functionsψ⁻¹, will give

|Y₂₂| ≤Cψ⁻¹t_hh

⎡

⎣

1≤j≤J−1

[[ψβh]]²_j+1

2 + [[ψβh]]²1

2 +γ[[ψβh]]²_J+1 2

⎤

⎦

1 2

≤C h⁻¹²ψ⁻¹t_hzh , (4.12b)

sinceγ <1. Similarly, sinceε <h we also have Y₂₃= √

ε J

j=0

[[σht_h]]_j+¹

2 −√

ε

σ_h⁺,[[t_h]]

⁰_h+√

ε(σ_h⁻t_h⁻)_J+¹

2 −√

ε(σ_h⁺t_h⁺)¹

2

= √ ε

[[σh]],t_h⁻

⁰_h ≤C h⁻¹²ψ⁻¹thzh , (4.12c)

after some simple manipulations and using in addition the inverse property of element bound- ary points to the interval for the functionσh. As a result, we can use (4.9) and get

|Y₂₂| + |Y₂₃| ≤C h⁻¹²ψ⁻¹t_hz_h ≤C h⁻¹²ψM(g_h)z_h

≤ 1

8ψM(gh)²+C h⁻¹zh². (4.13) Furthermore, it is followed from assumption (4.3) and inequality (4.9) that

|Y3| ≤Fψ⁻¹th ≤ 1

8ψM(gh)²+C F². (4.14) Till now we can collect up the above estimates from (4.11) to (4.14), and obtain the upper boundedness ofgh,th.

Finally we substitute the needed estimates into (4.8), and then complete the proof of this

lemma.

Lemma 4.2 Letβh ∈V_h is the solution of (4.1). If K is large enough, then there exists a bounding constant C independent of h,K andε⁻¹, such that

ψβh(T)²≤C

ψβh(0)²+ _T

0

F²(t)+h F²(t) dt

. (4.15)

Proof Take the test function v_h=Rh(ψ²z_h)in (4.1), which has been defined in Lemma3.5.

Then we will get F_h(Rh(ψ²z_h))+

βh,t,N^⊥_h(ψ²βh)

=

βh,t, ψ²βh

+B_h(z_h, ψ²z_h)−B_h(z_h,R^⊥_h(ψ²z_h))

≥ 1 2

d

dtψβh²+

1−C K⁻¹²

zh²−Cψβh², (4.16)

due to Lemmas3.4and3.5, as well as K ≥1. In what follows we will estimate successively two terms on the left-hand side.

Noticing the exact collocation of projectionNh, and using (3.8), we will get easily that Rh(ψ²zh)≤C

zh²+ ψβh²¹

2, (4.17)

where the bounding constant C >0 is independent of h,K and zh. From assumption (4.3) and relationship (4.17), Young’s inequality yields

|Fh(Rh(ψ²zh))| ≤FRh(ψ²zh)≤1 2

zh²+ ψβh²

+C F². (4.18) With the help of highest-frequency component and the definitions of Gauss–Radau pro- jections, we are able to get rid of the bad affections corresponding to the lower frequency