# A New Characteristic Nonconforming Mixed Finite Element Scheme for Convection-Dominated Diffusion Problem

Authors: Dongyang Shi; Qili Tang; Yadong Zhang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A characteristic nonconforming mixed finite element method (MFEM) is proposed for the convection-dominated diffusion problem based on a new mixed variational formulation. The optimal order error estimates for both the original variableu and the auxiliary
variable σ with respect to the space are obtained by employing some typical characters of the interpolation operator instead of the mixed (or expanded mixed) elliptic projection which is an indispensable tool in the traditional MFEM analysis. At last, we give some numerical results
to confirm the theoretical analysis.


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## Body

## 1. Introduction

Consider the following convection-dominated diffusion problem:(1)ut+a(x,y)·∇u-∇·(b(x,y)∇u)=f(x,y,t),(x,y,t)∈Ω×(0,T),u(x,y,t)=0,(x,y,t)∈∂Ω×(0,T),u(x,y,0)=u0(x,y),(x,y)∈Ω,
where Ω is a bounded polygonal domain in ℝ2 with Lipschitz continuous boundary ∂Ω,  J=(0,T],0<T<+∞. ∇ and  ∇· denote the gradient and the divergence operators, respectively.Model (1) has been widely used to describe the conduction of heat in fluid, the diffusion of soluble minerals or pollutants in ground water, the incompressible miscible displacement in porous media, and so on. The parameters appearing in (1) satisfy the following assumptions [1, 2]:(A1)
u denotes, for example, the concentration or saturation of soluble substances;(A2)
a(x,y)=(a1(x,y),a2(x,y)) represents Darcy velocity of mixed fluid, and f a source term;(A3)
b(x,y) is sufficiently smooth and there exist constants  b1 and  b2, such that
(2)0<b1≤b(x,y)≤b2<+∞,∀(x,y)∈Ω.It is well known that convection dominated-diffusion problem (1) often presents serious numerical difficulties. The standard numerical methods, such as finite difference method (FDM), FEM and MFEM, usually produce numerical diffusion along sharp fronts. In order to overcome this fatal defect, Douglas et al. [3] combined the method of characteristics with FE procedures so as to reduce the truncation error, and it allows us to use large time steps without lose of accuracy. Moreover, there have appeared many effective discretization schemes concentrating on the hyperbolic nature of the equation, for example, characteristic FD streamline diffusion method [4, 5], Eulerian-Lagrangian method [6, 7], characteristic-finite volume element method [2, 8, 9], characteristics-mixed covolume method [10, 11], the modified method of characteristic-Galerkin FE procedure [12], characteristic nonconforming FEM [13–15], characteristic MFEM [16–19] and expanded characteristic MFEM [1, 20], and so forth.As for the characteristic MFEM or expanded characteristic MFEM, the convergence rates ofu and σ in existing literature were suboptimal [11, 18, 21, 22] and the convergence analysis was valid only to the case of the lowest order MFE approximation [10, 17]. So far, to our best knowledge there are few studies on the optimal order error estimates except for [23], in which a family of characteristic MFEM with arbitrary degree of Raviart-Thomas-Nédélec space in [24, 25] for transient convection diffusion equations was studied.Recently, based on the low regularity requirement of the flux variable in practical problems, a new mixed variational form for second elliptic problem was proposed in [26]. It has two typical advantages: the flux space belongs to the square integrable space instead of the traditional H(div;Ω), which makes the choices of MFE spaces sufficiently simple and easy; the LBB condition is automatically satisfied when the gradient of approximation space for the original variable is included in approximation space for the flux variable. Motivated by this idea, this paper will construct a characteristic nonconforming MFE scheme for (1) with a new mixed variational formulation. Similar to the expanded characteristic MFEM, the coefficient b of (1) in this proposed scheme does not need to be inverted; therefore, it is also suitable for the case when b is small. By employing some distinct characters of the interpolation operators on the element instead of the mixed or expanded mixed elliptic projection used in [1, 17, 20] which is an indispensable tool in the traditional characteristic MFEM analysis, the O(h2) order error estimate in L2-norm for original variable u, which is one order higher than [1, 20] and half order higher than [18], is derived, and the optimal error estimates with order O(h) for auxiliary variable σ in L2-norm and for u in broken H1-norm are obtained, respectively. It seems that the result for u in broken H1-norm has never been seen in the existing literature by making full use of the high-accuracy estimates of the lowest order Raviart-Thomas element proved by the technique of integral identities in [27] and the special properties of nonconforming EQ1rot element (see Lemma 1 below).The paper is organized as follows. Section2 is devoted to the introduction of the nonconforming FE approximation spaces and their corresponding interpolation operators. In Section 3, we will give the construction of the new characteristic nonconforming MFE scheme and two important lemmas, and the existence and uniqueness of the discrete scheme solution will be proved. In Section 4, the convergence analysis and optimal error estimates for both the original variable u and the flux variable σ are obtained. In Section 5, some numerical results are provided to illustrate the effectiveness of our proposed method.Throughout this paper,C denotes a generic positive constant independent of the mesh parameters  h and  Δt with respect to domain  Ω  and time  t.


## 2. Construction of Nonconforming MFEs

As in [28], we frequently employ the space L2(Ω) of square integrable functions with scalar product and norm
(3)(u,v)=(u,v)L2(Ω)=(∫Ωuvdxdy)1/2,∥v∥=∥v∥L2(Ω)=(∫Ωv2dxdy)1/2.
We also employ the Sobolev space  Hm(Ω),m≥1, of functions v such that Dβv∈L2(Ω) for all  |β|≤m,  equipped with the norm and seminorm
(4)∥v∥m,Ω=∥v∥Hm(Ω)=(∑|β|≤m∥Dβv∥2)1/2,|v|m,Ω=|v|Hm(Ω)=(∑|β|=m∥Dβv∥2)1/2.
The space H01(Ω) denotes the closure of the set of infinitely differentiable functions with compact supports in Ω. For any Sobolev space Y, Lp(0,T;Y) is the space of measurable Y-valued functions Φ of t∈(0,T), such that  ∫0T‍∥Φ(·,t)∥Ypdt<∞  if  1≤p<∞, or such that   esssup0<t<T∥Φ(·,t)∥Y<∞  if  p=∞.We now introduce the nonconforming MFE space described in [29] for and summarize it as follows.LetΩ⊂ℝ2 be a polygon domain with edges parallel to the coordinate axes on xy plane, and let  Th be a rectangular subdivision of Ω satisfying the regular condition [30]. For a given element e∈Th, denote the barycenter of element e by (xe,ye), denote the length of edges parallel to x-axis and y-axis by 2hxe and 2hye, respectively, he=maxe∈Th{hxe,hye},h=maxe∈Thhe.Lete^=[-1,1]×[-1,1] be the reference element on x^y^ plane and four vertices d^1=(-1,-1), d^2=(1,-1),  d^3=(1,1), and d^4=(-1,1), the four edges l^1=d^1d^2¯, l^2=d^2d^3¯, l^3=d^3d^4¯, and l^4=d^4d^1¯. Then there exists an affine mapping Fe:e^→e  as
(5)x=xe+hxex^,y=ye+hyey^.
Define the FE spaces  (e^,Pi^,∑‍^i),  (i=1,2,3) by
(6)∑‍^1={v^1,v^2,v^3,v^4,v^5},P^1=span{1,x^,y^,ϕ(x^),ϕ(y^)},∑‍^2={p^1,p^2},P^2=span{1,x^},∑‍^3={q^1,q^2},P^3=span{1,y^},
where v^i=(1/|l^i|)∫l^i‍v^ds^,(i=1,2,3,4), v^5=(1/|e^|)∫e^‍v^dx^dy^, ϕ(t)=(1/2)(3t2-1), p^i=(1/|l^2i|) ∫l^2i‍p^ds^, q^i=(1/|l^2i-1|)∫l^2i-1‍q^ds^,(i=1,2).The interpolation operators one^ are defined as follows:
(7)Π^1:v^∈H1(e^)⟶Π^1v^∈P^1,∫l^i(Π^1v^-v^)ds^=0,(i=1,2,3,4),∫e^(Π^1v^-v^)dx^dy^=0,Π^2:p^∈H1(e^)⟶Π^2p^∈P^2,∫l^2i(Π^2p^-p^)ds^=0,(i=1,2),Π^3:q^∈H1(e^)⟶Π^3q^∈P^3,∫l^2i-1(Π^3q^-q^)ds^=0,(i=1,2).
Then the associated nonconforming  EQ1rot element space Mh [29] and lowest order Raviart-Thomas element space Vh [25, 27] are defined as
(8)Mh={∫F‍vh:vh∣e=v^∘Fe-1,v^∈P^1,∫F‍[vh]ds=0,F⊂∂e},Vh={P^3wh=(wh1,wh2):wh∣e=(w^1∘Fe-1,w^2∘Fe-1),w^=(w^1,w^2)∈P^2×P^3},
respectively, where [φ] represents the jump value of φ across the boundary F, and [φ]=φ if  F⊂∂Ω.Similarly, the interpolation operatorsπh1  and  πh2 are defined as
(9)πh1:H1(Ω)⟶Mh,πh1|e=πe1,πe1v=(Π^1v^)∘Fe-1,∀v∈H1(Ω),πh2:(H1(Ω))2⟶Vh,πh2|e=πe2,πe2w=((Π^2w^1)∘Fe-1,(Π^3w^2)∘Fe-1),Πe2w=∀w=(w1,w2)∈(H1(Ω))2.


## 3. New Characteristic Nonconforming MFE Scheme and Two Lemmas

Letψ(x,y)=(1+|a(x,y)|2)1/2  and τ=τ(x,y)  be the characteristic direction associated with  ut+ a(x,y)·∇u,  such that
(10)∂∂τ=1ψ(x,y)∂∂t+a(x,y)ψ(x,y)·∇.Then (1) can be put in the following system:
(11)ψ(x,y)∂u∂τ-∇·(b(x,y)∇u)=f(x,y,t),ψ(x,y)∂u∂τ∀(x,y,t)∈Ω×(0,T],u(x,y,t)=0,(x,y,t)∈∂Ω×(0,T],u(x,y,0)=u0(x,y),(x,y)∈Ω.By introducingσ=-b(x,y)∇u and using Green's formula, we obtain the new characteristic mixed form of (11). Find (u,σ):(0,T]→H01(Ω)×(L2(Ω))2, such that
(12)(ψ(x,y)∂u∂τ,v)-(σ,∇v)=(f(x,y,t),v)∀v∈H01(Ω),(σ,w)+(b(x,y)∇u,w)=0,∀w∈(L2(Ω))2.LetΔt>0,N=T/Δt∈ℤ,  tn=nΔt, and ϕn=ϕ(x,y,tn). When solving  uhn+1, we would like to make the scheme as implicit as possible by using of the characteristic vector τ. Denote X=(x,y)∈Ω and
(13)X-=X-a(x,y)Δt,
similar to [1, 3], and then we have the following approximation:
(14)ψ(x,y)∂u∂τ|tn≈ψ(x,y)u(X,tn)-u(X-,tn-1)(X-X-)2+(Δt)2=u(X,tn)-u(X-,tn-1)Δt=un-u-n-1Δt.This leads to the following characteristic nonconforming MFE scheme. Find(uh,σh):{t0,t1,…, tN}→Mh×Vh, such that (15a)(uhn-u-hn-1Δt,vh)-(σhn,∇vh)h=(fn,vh),∀vh∈Mh,(15b)(σhn,wh)+(b∇uhn,wh)h=0,∀wh∈Vh,(15c)uh0=πh1u0(x,y),σh0=πh2(b∇u0(x,y)),∀(x,y)∈Ω,where u-hn=uh(X-,tn), (u,v)h=∑e∈Th∫euvdxdy. Generally speaking, u-hn-1(n=2,…,N) are not node values and should be derived by interpolation formulas on uhn-1.Remark 1.
In [1], the expanded characteristic MFE scheme was presented by introducing two new auxiliary variables which avoided the inversion of the coefficient b when b is small. The new mixed schemes (15a), (15b), and (15c) not only keep the advantage of expanded characteristic MFE scheme, but also donot need to solve three variables.Now, we prove the existence and uniqueness of the solution of (15a), (15b), and (15c).Theorem 1.
Under assumption (A3), there exists a unique solution(uh,σh)∈Mh×Vh to the schemes (15a), (15b), and (15c).Proof.
The linear system generated by (15a), (15b), and (15c) is square, so the existence of the solution is implied by its uniqueness. From (15a), (15b), and (15c), we have
(16)(uhnΔt,vh)-(σhn,∇vh)h=(u-hn-1Δt,vh)+(fn,vh),∀vh∈Mh,(σhn,wh)+(b∇uhn,wh)h=0,∀wh∈Vh.
Letuhn and f be zero, and  thus  u-hn is zero too; taking vh=uhn,wh=(1/b)σhn in (16) and adding them together, we have
(17)1Δt∥uhn∥2+(1bσhn,σhn)=0.
Thus assumption (A3) implies that uhn=σhn=0. The proof is complete.To get error estimates, we state the following two important lemmas.Lemma 1 (see [27, 29, 31]).
Assume thatu∈H1(Ω),p∈(H2(Ω))2, for all vh∈Mh,wh∈Vh,  and then there hold
(18)(∇(u-πh1u),∇vh)h=0,(∇(u-πh1u),wh)h=0,(19)(p-πh2p,wh)≤Ch2|p|2,Ω∥wh∥,(20)|∑e∈Th∫∂epvh·nds|≤Ch2|p|2,Ω∥vh∥1,h,
where ∥·∥1,h=(∑e∈Th|·|1,e)1/2 is a norm on Mh, and n denotes the outward unit normal vector on ∂e.Lemma 2 (see [1, 3]).
Letφ∈L2(Ω), and φ-=φ(X-g(X)Δt), where function g and its gradient  ∇g are bounded, then
(21)∥φ-φ-∥-1≤C∥φ∥Δt,
where ∥φ∥-1=supϕ∈H1(Ω)((φ,ϕ)/∥ϕ∥1,Ω).


## 4. Convergence Analysis and Optimal Order Error Estimates

In this section, we aim to analyze the convergence analysis and error estimates of characteristic nonconforming MFEM. In order to do this, let(22)uh-u=uh-πh1u+πh1u-u=e+ρ,σh-σ=σh-πh2σ+πh2σ-σ=ξ+η.
Taking t=tn in (12) yields(23a)(ψ∂un∂τ,vh)-(σn,∇vh)h+∑e∈Th∫∂eσnvh·nds=(fn,vh),(ψ∂un∂τ,vh)-(σn,∇vh)h+∑e∈Th∫∂eσnvh∀vh∈Mh,(23b)(σn,wh)+(b∇un,wh)h=0,∀wh∈Vh.From (23a), (23b), (15a), (15b), and (15c) we get(24a)(en-e-n-1Δt,vh)-(ξn,∇vh)h=(ψ∂un∂τ-un-u-n-1Δt,vh)-(ρn-ρ-n-1Δt,vh)+(ηn,∇vh)h+∑e∈Th∫∂eσnvh·nds,∀vh∈Mh,(24b)(ξn,wh)+(b∇en,wh)h=-(ηn,wh)-(b∇ρn,wh)h,(ξn,wh)+(b∇en,wh)h=-(ηn,wh)∀wh∈Vh.We are now in a position to prove the optimal order error estimates.Theorem 2.
Let(u,σ) and (uhn,σhn) be the solutions of (12), (15a), (15b), and (15c), respectively, (∂2u/∂τ2)∈L2(0,T;L2(Ω)),ut∈L2(0,T;H2(Ω)),u∈L∞(0,T;H2(Ω)),σ∈L∞(0,T;H2(Ω)) and assume that  Δt=O(h2). Then under assumption (A3), we have
(25)max0≤n≤N∥(uh-u)(tn)∥1,h≤C(Δt+h),(26)max0≤n≤N∥(uh-u)(tn)∥≤C(Δt+h2),(27)max0≤n≤N∥(σh-σ)(tn)∥≤C(Δt+h).Proof.
Takingvh=en in (24a) and wh=∇en in (24b), and adding them, we have
(28)(en-e-n-1Δt,en)+(b∇en,∇en)h=(ψ∂un∂τ-un-u-n-1Δt,en)-(ρn-ρn-1Δt,en)-(ρn-1-ρ-n-1Δt,en)+∑e∈Th∫∂eσnen·nds-(b∇ρn,∇en)h=∑i=15(Err)i.
On the one hand, we consider the right hand of (28).
Using the method similar to [3], we have
(29)(Err)1≤C∥ψ∂un∂τ-un-u-n-1Δt∥2+ε12∥en∥2≤CΔt∥∂2u∂τ2∥L2(tn-1,tn;L2(Ω))2+ε12∥en∥2.(Err)2 can be estimated as
(30)|(Err)2|≤1Δt(∫Ω(∫tn-1tnρtds)2dxdy)1/2∥en∥≤1Δt(∫Ω∫tn-1tn‍ρt2dsdxdy)1/2∥en∥≤CΔt∫tn-1tn∥ρt∥2ds+ε12∥en∥2≤Ch4Δt∫tn-1tn∥ut∥2,Ω2ds+ε12∥en∥2.
By Lemma 2, we obtain
(31)|(Err)3|≤1Δt∥ρn-1-ρ-n-1∥-1∥en∥1,h≤C∥ρn-1∥2+b16∥en∥1,h2≤Ch4∥un-1∥2,Ω2+b16∥en∥1,h2.
It follows from Lemma 1 that
(32)|(Err)4|≤Ch4∥σn∥2,Ω2+b16∥en∥1,h2.
Let b-=(1/|e|)∫eb(x,y)dxdy. By Lemma 1, we have
(33)|(Err)5|=|-((b-b-)∇ρn,∇en)h|≤Ch|b|W1,∞(Ω)∥ρn∥1,h∥en∥1,h≤Ch4∥un∥2,Ω2+b16∥en∥1,h2.
On the other hand, the left hand of (28) can be bounded by
(34)(en-e-n-1Δt,en)+(b∇en,∇en)h≥12Δt((en,en)-(e-n-1,e-n-1))+b1∥en∥1,h2≥12Δt(∥en∥2-(1+CΔt)∥en-1∥2)+b1∥en∥1,h2,
where the inequality ∥e-n-1∥2≤(1+CΔt)∥en-1∥2 proved in [3] is used in the last step.
Combining (29)–(34) with (28) gives
(35)12Δt(∥en∥2-∥en-1∥2)+b1∥en∥1,h2≤C(Δt∥∂2u∂τ2∥L2(tn-1,tn;L2(Ω))2+h4Δt∫tn-1tn∥ut∥2,Ω2ds+h4(∥un-1∥2,Ω2+∥un∥2,Ω2+∥σn∥2,Ω2)∥∂2u∂τ2∥L2(tn-1,tn;L2(Ω))2)+ε1∥en∥2+C∥en-1∥2+b12∥en∥1,h.
Taking 1-2Δtε1>0, multiplying (35) by 2Δt, summing over from i=1 to i=n, and noticing that  e0=0,  we obtain
(36)∥en∥2+Δt∑i=1n∥ei∥1,h2≤C((Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2+h4∫0tn∥ut∥2,Ω2ds+Δth4∑i=1n(∥ui∥2,Ω2+∥σi∥2,Ω2))+C∑i=1n-1∥ei∥2.
It follows from discrete Gronwall’s lemma that
(37)∥en∥2+Δt∑i=1n∥ei∥1,h2≤C((Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2≤C+h4(∥σ∥L∞(0,tn;(H2(Ω))2)2∥ut∥L2(0,tn;H2(Ω))2+∥u∥L∞(0,tn;H2(Ω))2+∥σ∥L∞(0,tn;(H2(Ω))2)2)∥∂2u∂τ2∥L2(0,tn;L2(Ω))2).
From (37) we get the optimal order error estimate of ∥en∥ rather than ∥en∥1,h. So we start to reestimate ∥en∥1,h in the following manner and derive the estimation of ∥ξn∥  simultaneously.
Firstly, choosingvh=((en-en-1)/Δt) in (24a) and wh=∇((en-en-1)/Δt) in (24b), and adding them, we have
(38)(en-e-n-1Δt,en-en-1Δt)+(b∇en,∇en-en-1Δt)h=(ψ∂un∂τ-un-u-n-1Δt,en-en-1Δt)-(ρn-ρn-1Δt,en-en-1Δt)-(ρn-1-ρ-n-1Δt,en-en-1Δt)+∑e∈Th∫∂eσnen-en-1Δt·nds-(b∇ρn,∇en-en-1Δt)h=∑i=15‍(Err)i′.
The left hand can be estimated as
(39)(en-e-n-1Δt,en-en-1Δt)+(b∇en,∇en-en-1Δt)h≥∥en-en-1Δt∥2+12Δt[(b∇en,∇en)-(b∇en-1,∇en-1)]+(en-1-e-n-1Δt,en-en-1Δt),
and (Err)i′,(i=1,2,3,4,5) can be bounded by
(40)|(Err)1′|≤CΔt∥∂2u∂τ2∥L2(tn-1,tn;L2(Ω))2+14∥en-en-1Δt∥2,|(Err)2′|≤Ch4Δt∫tn-1tn∥ut∥2,Ω2ds+14∥en-en-1Δt∥2,|(Err)3′|≤Ch4Δt∥un-1∥2,Ω2+ε3Δt∥en-en-1Δt∥1,h2,|(Err)4′|≤Ch4Δt∥σn∥2,Ω2+ε3Δt∥en-en-1Δt∥1,h2,|(Err)5′|≤Ch4Δt∥un∥2,Ω2+ε3Δt∥en-en-1Δt∥1,h2.
From (38)–(40), we get
(41)12∥en-en-1Δt∥2+12Δt[(b∇en,∇en)h-(b∇en-1,∇en-1)h]≤C[Δt∥∂2u∂τ2∥L2(tn-1,tn;L2(Ω))2+h4Δt(∫tn-1tn∥ut∥2,Ω2ds+∥un∥2,Ω2+∥un-1∥2,Ω2+∥σn∥2,Ω2∥∂2u∂τ2∥L2(tn-1,tn;L2(Ω))2)]+εΔt∥en-en-1Δt∥1,h2+(en-1-e-n-1Δt,en-en-1Δt).
Multiplying (41) by 2Δt and summing over in time from i=1 to i=n yield
(42)Δt∥en-en-1Δt∥2+b1∥en∥1,h2≤C[(Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2+h4∥ut∥L2(0,tn;H2(Ω))2+h4∑i=1n(∥ui∥2,Ω2+∥σi∥2,Ω2)]+ε(Δt)2∑i=1n∥ei-ei-1Δt∥1,h2+∑i=1n(ei-1-e-i-1Δt,ei-ei-1).
Secondly, we take Δt→0 and Δt must approach zero in such a way that  Δt  and  h  satisfy
(43)Δt=O(h2),
and by inverse inequality, we have
(44)(Δt)2∑i=1n∥ei-ei-1Δt∥1,h2≤CΔt∑i=1n∥ei-ei-1Δt∥2.
At the same time, using Lemma 2, we obtain
(45)∑i=1n(ei-1-e-i-1Δt,ei-ei-1)=(en-1-e-n-1Δt,en)+∑i=1n-1(ei-1-ei-(e-i-1-e-i)Δt,ei)≤C∥en-1∥∥en∥1,h+∑i=1n-1∥ei-ei-1∥∥ei∥1,h≤C∥en-1∥2+b12∥en∥1,h2+Δt∑i=1n-1∥ei-ei-1Δt∥2+CΔt∑i=1n-1∥ei∥1,h2.
From (42)–(45), taking suitable small ε such that  1-εC>0, we have
(46)Δt∥en-en-1Δt∥2+∥en∥1,h2≤C[(Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2+h4∥ut∥L2(0,tn;H2(Ω))2+h4∑i=1n(∥ui∥2,Ω2+∥σi∥2,Ω2)]+∥en-1∥2+CΔt∑i=1n-1∥ei-ei-1Δt∥2+CΔt∑i=1n-1∥ei∥1,h2.
Finally, applying discrete Gronwall’s lemma yields
(47)∥en∥1,h2≤C[(Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2+h4∥ut∥L2(0,tn;H2(Ω))2∥en∥1,h2≤+h2(∥u∥L∞(0,tn;H2(Ω))2+∥σ∥L∞(0,tn;H2(Ω))2)∥∂2u∂τ2∥L2(0,tn;L2(Ω))2].
In order to derive (27), set wh=ξn in (24b) and employ Lemma 1 and assumption (A3) to give
(48)∥ξn∥2=-(b∇en,ξn)h-(ηn,ξn)-(b∇ρn,ξn)h≤C(∥en∥1,h2+h4∥σn∥2,Ω)-((b-b-)∇ρn,ξn)h+14∥ξn∥2≤C(∥en∥1,h2+h4(∥σn∥2,Ω+∥un∥2,Ω))+12∥ξn∥2.
Combining (47) with (48) yields
(49)∥ξn∥2≤C[(Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2+h4∥ut∥L2(0,tn;H2(Ω))2+h2(|u|L∞(0,tn;H2(Ω))2+∥σ∥L∞(0,tn;H2(Ω))2)∥∂2u∂τ2∥L2(0,tn;L2(Ω))2].
By using of interpolation theory and the triangle inequality, (37), (47), and (49) lead to (25), (26), and (27), respectively, which are the desired results.Remark 2.
From (37), we have
(50)Δt∑i=1n‍∥ei∥1,h2=Δt∑i=1n∥(πh1u-uh)i∥1,h2≤C((Δt)2∥∂2u∂τ2∥L2(0,tn;L2(Ω))2≤C+h4(∥ut∥L2(0,tn;H2(Ω))2+∥u∥L∞(0,tn;H2(Ω))2+∥σ∥L∞(0,tn;(H2(Ω))2)2)∥∂2u∂τ2∥L2(0,tn;L2(Ω))2).
This byproduct can be regarded as the superclose result between πh1u and uh in mean broken H1-norm. It seems that both (25) and (50) have never been seen in the existing studies. At the same time, by employing the new characteristic nonconforming MFE scheme, we can also obtain the same error estimate of (27) as traditional characteristic MFEM [10].Remark 3.
From the analysis of Theorem2 in this paper, we may see that Lemma 1 is the key result leading to the successful optimal order error estimations. If we want to get higher order accuracy, similar to Lemma 1, the nonconforming finite elements for approximating u should also possess a very special property, that is, the consistency error estimates with O(h2) order, and satisfy (18). For the famous nonconforming Wilson element [32] whose shape function is span{1,x,y,x2,y2}, by a counter-example, it has been proven in [32] that its consistency error estimate is of O(h) order and cannot be improved any more. For the rotated bilinear Q1 element [33] whose shape function is span{1,x,y,x2-y2}, although its consistency error with O(h2) order and (∇(u-πh1u),∇vh)h=0 on square meshes is satisfied, the second term of (18) is not valid. Thus when they are applied to (1) on new characteristic mixed finite element scheme, up to now, the optimal order error estimates of (25), (26), and (27) cannot be obtained directly.


## 5. Numerical Example

In order to verify our theoretical analysis in previous sections, we consider the convection-dominated diffusion problem (1) as follows:
(51)ut+ux+uy-10-4(uxx+uyy)=f(x,y,t),(x,y,t)∈Ω×(0,T),u(x,y,t)=0,(x,y,t)∈∂Ω×(0,T),u(x,y,0)=u0(x,y),(x,y)∈Ω
with Ω=[0,1]×[0,1],a(x,y)=(1,1), and b(x,y)=10-4.The right hand termf(x,y,t) is taken such that u=e-tsin(πx)sin(2πy),σ=-10-4e-t(πcos(πx) sin(2πy),2πsin(πx)cos(2πy)) are the exact solutions.We first divide the domainΩ into m and n equal intervals along x-axis and y-axis and the numerical results at different times are listed in Tables 1, 2, and 3 and pictured in Figures 1, 2, 3, and 4, respectively. (uh,ph) denotes the characteristic nonconforming MFE solution of the problem (15a), (15b), and (15c). Δt represents the time step and the experiment is done with Δt=h2. α stands for the convergence order.Table 1
Numerical results of∥u-uh∥1,h.
m
×
n
t
=
0.2
α
t
=
0.3
α
t
=
0.4
α
8
×
8
0.75277
/
0.75017
/
0.66433
/
16
×
16
0.42984
0.81
0.41849
0.84
0.35474
0.91
32
×
32
0.21758
0.99
0.21412
0.97
0.17552
1.02
m
×
n
t
=
0.5
α
t
=
0.8
α
t
=
0.9
α
8
×
8
0.55291
/
0.42211
/
0.40937
/
16
×
16
0.29234
0.92
0.23117
0.87
0.21120
0.96
32
×
32
0.14466
1.02
0.10807
1.10
0.09343
1.18Table 2
Numerical results of∥u-uh∥.
m
×
n
t
=
0.4
α
t
=
0.5
α
t
=
0.7
α
8
×
8
0.0298190
/
0.0276370
/
0.0223240
/
16
×
16
0.0073087
2.03
0.0062445
2.15
0.0048038
2.22
32
×
32
0.0020769
1.82
0.0017926
1.80
0.0013309
1.85
m
×
n
t
=
0.8
α
t
=
0.9
α
t
=
1.0
α
8
×
8
0.0198730
/
0.0175900
/
0.0154090
/
16
×
16
0.0044472
2.16
0.0041982
2.07
0.0039150
1.98
32
×
32
0.0011894
1.90
0.0010738
1.97
0.0009466
2.05Table 3
Numerical results of∥σ-σh∥.
m
×
n
t
=
0.1
α
t
=
0.4
α
t
=
0.5
α
8
×
8
4.9528
e
-
005
/
4.2661
e
-
005
/
3.8292
e
-
005
/
16
×
16
2.3945
e
-
005
1.05
1.8843
e
-
005
1.18
1.6806
e
-
005
1.19
32
×
32
1.1749
e
-
005
1.03
9.0029
e
-
006
1.07
8.0521
e
-
006
1.06
m
×
n
t
=
0.7
α
t
=
0.8
α
t
=
0.9
α
8
×
8
3.0714
e
-
005
/
2.7735
e
-
005
/
2.524
e
-
005
/
16
×
16
1.3326
e
-
005
1.20
1.224
e
-
005
1.18
1.1443
e
-
005
1.14
32
×
32
6.455
e
-
006
1.05
5.8353
e
-
006
1.07
5.3751
e
-
006
1.09Figure 1
Errors att=0.4.Figure 2
Errors att=0.5.Figure 3
Errors att=0.8.Figure 4
Errors att=0.9.It can be seen from the above Tables1, 2, and 3 that ∥u-uh∥1,h and ∥σ-σh∥ are convergent at optimal rate of O(h) and ∥u-uh∥ is convergent at optimal rate of O(h2), respectively, which coincide with our theoretical investigation in Section 4.



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*Source: 951692-2013-04-17.xml*