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THE
hp
VERSION OF EULERIAN-LAGRANGIAN MIXED
DISCONTINUOUS FINITE ELEMENT METHODS
FOR ADVECTION-DIFFUSION PROBLEMS
HONGSEN CHEN, ZHANGXIN CHEN, and BAOYAN LI
Received 26 December 2001
We study thehpversion of three families of Eulerian-Lagrangian mixed discon-tinuous finite element (MDFE) methods for the numerical solution of advection-diffusion problems. These methods are based on a space-time mixed formulation of the advection-diffusion problems. In space, they use discontinuous finite ele-ments, and in time they approximately follow the Lagrangian flow paths (i.e., the hyperbolic part of the problems). Boundary conditions are incorporated in a natu-ral and mass conservative manner. In fact, these methods are locally conservative. The analysis of this paper focuses on advection-diffusion problems in one space dimension. Error estimates are explicitly obtained in the grid sizeh, the polyno-mial degreep, and the solution regularity; arbitrary space grids and polynomial degree are allowed. These estimates are asymptotically optimal in bothhandp
for some of these methods. Numerical results to show convergence rates inhand
p of the Eulerian-Lagrangian MDFE methods are presented. They are in a good agreement with the theory.
2000 Mathematics Subject Classification: 35K60, 35K65, 65N30, 65N22.
1. Introduction. The development of discontinuous Galerkin finite element methods for the numerical solution of partial differential equations dates back to the early 1970s. They were first introduced in [34] for the neutron transport equation (a first-order linear differential equation). Their extensions, stability, and convergence analyses were later carried out by many researchers [19,26, 27,28,29,35].
A slightly different discontinuous Galerkin method has recently been intro-duced in [32]. Their method is different from the global element method in that the sign in front of the so-called symmetric term was altered. The disad-vantage of this method is that the bilinear form is nonsymmetric even for a symmetric differential problem. Also, only suboptimal error estimates in the
L2-norm for numerical solutions can be obtained. This method and its
penal-ized versions have been studied in [36]. For the relationship between all the existing discontinuous Galerkin methods, we refer to [13].
There have been a lot of applications of the above two families of discon-tinuous Galerkin methods and their stabilized versions to practical problems. Among them are the applications to fluid flows in porous media [11,17], to the Euler equations [7], to nonlinear acoustic waves [30], and to semiconductor device modeling [15,16], for example.
Discontinuous finite element methods in mixed form were utilized in [15, 16] and also developed in [6]. While the lowest-order Raviart-Thomas space [33] was used in [15, 16], Lagrange multipliers on interior boundaries were exploited to relax the continuity of normal derivatives of functions in the vector space, so all finite elements are discontinuous. These methods in mixed form have been recently studied in [20] for the convection-diffusion equations and in [8] for the Laplace equation.
The objective of this paper is to study thehpversion of the three families of Eulerian-Lagrangian mixed discontinuous finite element (MDFE) methods and their equivalent Galerkin versions for the numerical solution of advection-diffusion problems. These methods are based on a space-time mixed formu-lation of the advection-diffusion problems. In space, they use discontinuous finite elements, and in time, they approximately follow the Lagrangian flow paths (i.e., the hyperbolic part of the problems). Boundary conditions are in-corporated in a natural and mass conservative manner. In fact, these meth-ods are locally conservative. A characteristic treatment to handle the advec-tion term in time is similar to those exploited in [1,10]; it is known that this technique is suitable for advection-dominated problems. Also, the stationary version of the MDFE methods (i.e., as the MDFE methods applied to elliptic problems) has been studied in [12]. The first family (see (3.1)) was developed in [13,14] and extends the mixed discontinuous methods originally proposed in [6,15,16,20], the second family (see (4.2)) is a generalization of the local discontinuous Galerkin method in [9] (which is a particular case of the local discontinuous method introduced in [20]), and the third family (see (5.1)) is a modification of the second family.
is obtained for oddpand an optimal rate is derived for evenp, see Theorems 3.2and3.3. For the second and third families, the optimal rate of convergence can be achieved for certain boundary conditions or by choosing appropriately a parameter at the boundary of the domain, see Theorems4.2and5.2. Numerical results to show convergence rates of the Eulerian-Lagrangian MDFE methods in p and h are presented. They are in a good agreement with the obtained theoretical results.
As observed in [13,14], when discontinuous finite element methods are de-fined in a mixed form, they not only preserve good features of these methods, but also have some advantages over the classical Galerkin form such as they are more stable and more accurate (in space). While an auxiliary variable is introduced in the mixed formulation, the MDFE methods can be reduced to the Galerkin version in one of the two variables because of their local property and can be implemented (if desired) in nonmixed form [13,14].
The rest of the paper is outlined as follows. InSection 2, we describe the con-tinuous problem and its mixed weak formulation. The first, second, and third families of the Eulerian-Lagrangian MDFE methods are defined and discussed in Sections3,4, and5, respectively. The proof of the convergence results is presented inSection 6. Some generalizations of these methods are mentioned inSection 7. Numerical experiments for showing convergence rates in bothp
andhare presented inSection 8.
2. The continuous problem. We consider the advection-diffusion equation foruon a bounded intervalI=(c, d)with boundary∂I=ΓD∪ΓN,ΓD∩ΓN= ∅:
∂(φu)
∂t +
∂ ∂x
bu−a∂u ∂x
=f inI×J, u=gD, onΓD×J,
bu−a∂u ∂x
νI=gN, onΓN×J,
u(x,0)=u0(x) inI,
(2.1)
whereJ=(0, T ] (T >0), a(x, t),b(x, t), φ(x, t)∈L∞(I),gD(x, t)∈L∞(ΓD),
gN(x, t)∈L∞(ΓN),f (x, t)∈L2(I)(for eacht∈J), andu0(x)∈L2(I)are given functions (the standard Sobolev spacesHk(I)=Wk,2(I)with the usual norms
are used in this paper), andνIis the outer unit normal to∂I(while we call it a normal, it is, in fact, a scalar in one dimension; i.e., it is 1 or−1). To define the mixed weak formulation, we rewrite this equation as follows:
∂(φu) ∂t +
∂
∂x(bu−σ )=f inI×J,
σ=a∂u
∂x inI×J, u=gD, onΓD×J,
(bu−σ )νI=gN, onΓN×J,
u(x,0)=u0(x) inI.
• • • • •
xi
x0 xM
←ν ←ν ←ν ←ν ν→
Figure2.1. An illustration of the unit normalν.
Namely, an auxiliary variableσ is introduced, which has a physical meaning in applications such as the electric field in semiconductor modeling [15,16] or the velocity field in petroleum simulation [23,25], as mentioned in the in-troduction. In the next four sections, we study the case where the following assumption holds:
a∗≥a(x, t)≥a∗>0 inI×J, (2.3)
witha∗ anda∗ being constants. The situation without this assumption will be addressed inSection 7. Also, to demonstrate the idea of the proposed ap-proximation scheme, we consider the case whereφand bare constants; the nonconstant case will be addressed inSection 7as well.
Forh >0, let Ih be a partition ofI into subintervals Ii =(xi−1, xi), i= 1, . . . , M, with lengthhi=xi−xi−1≤h. LetᏵohdenote the set of all interior nodes ofIh, Ᏽb
hthe set of the end points on∂I, andᏵh=Ᏽoh∪Ᏽbh. We tacitly assume thatᏵoh= ∅.
Forl≥0, define
HlIh
=v∈L2(I):v|Ii∈H lI
i
, i=1, . . . , M. (2.4)
With eachxi∈Ᏽh, we associate a unit normalν. Forxi∈Ᏽbh,ν=νxi is just the outer unit normal to∂I; that is, for the left end,ν= −1 and for the right end,ν=1. Forxi∈Ᏽoh, it is chosen pointing to the element with lower index (seeFigure 2.1); that is,ν= −1 at all interior points. This is just for notational convenience; other choices are possible.
Forv∈Hl(I
h)withl >1/2, we define its average and jump atxi∈Ᏽohas follows (seeFigure 2.2):
{v}xi= 1 2
v|Iixi+v|Ii+1xi, [v]xi=v|Ii+1xi−v|Iixi.
(2.5)
Atxi∈Ᏽbh, the following convention is used (from insideI):
{v}xi
=vxi
, [v]xi
=
vxi
, ifxi∈ΓD,
0, ifxi∈ΓN.
(2.6)
• • •
xi
Ii Ii+1
Figure2.2. An illustration for the jump definition.
vn=v(·, tn)and
∆t= max
1≤n≤ᏺ∆t
n. (2.7)
The origin of our approximation scheme can be seen by considering (2.2) in a space-time framework. For anyx∈I and two times 0≤tn−1< tn≤T, the hyperbolic part of problem (2.1),φ∂u/∂t+b∂u/∂x, defines the characteristic ˇ
xn(x, t)along the interstitial velocityϕ=b/φ(it emanates backward fromx attn, seeFigure 2.3):
ˇ
xn(x, t)=x−ϕ
tn−t, t∈ˇt(x), tn, (2.8)
where ˇt(x)=tn−1if ˇx
n(x, t)does not backtrack to the boundary∂Ifort∈
[tn−1, tn]; ˇt(x)∈(tn−1, tn]is the time instant when ˇx
n(x, t)intersects∂I, that is, ˇxn(x,ˇt(x))∈∂I, otherwise. Ifb >0, the characteristic at the right boundary
(x=d, t∈Jn)is defined by
ˇ
xn(d, t)=d−ϕ(t−θ), θ∈
tn−1, t. (2.9)
Similarly, we can modify the characteristic at the left boundary(x=c, t∈Jn) ifb <0.
For some element Ii∈Ih, let ˇIi(t)indicate the traceback of Ii to timet,
t∈Jn:
ˇ
Ii(t)=
x∈I:x=xˇn(y, t)for somey∈Ii
. (2.10)
Also, letIinbe the space-time region that follows the characteristics:
Ini =
(x, t)∈I×J:t∈Jn, x∈ˇIi(t). (2.11)
For an element on either the boundary(x=c, t∈Jn)(ifb <0) or the boundary
(x=d, t∈Jn)(ifb >0), an analogous definition can be given (seeFigure 2.3). Let(·,·)S denote theL2(S)inner product (we omitS ifS=I). Multiplying the first equation of (2.2) byv∈H1(In
i), integrating the resulting equation on
In
i, and using Green’s formula, we see that
φun, vn Ii−
φun−1, vn−1 ˇ
Ii(tn−1)+
Jn
σ ,∂v ∂x
ˇ
Ii(t)
−σ νˇI i(t), v
∂ˇIi(t)
dt
=(f , v)Ini −
ubνI, v
∂Ini∩(∂I×Jn)+
u, φ∂v ∂t +b
∂v ∂x
Ini
,
tn
tn−1
ˇ
t1
x0 xi−1 xi xi+1 xM
ˇ
xn(xi, tn−1)
ˇ
tM+1≡xM+1 ˇ
xn(xi, t) ν
Figure2.3. An illustration of characteristics.
where we use the fact that(b, φ)·ν=0 on the space-time sides(∂Iin∩Jn)\
∂Iin∩(∂I×Jn)
andνˇIi(t)is the unit normal outward to ˇIi(t).
We denote the inverse of ˇxn(·, t)by ˆxn(·, t). For a functionv(x, t), ift∈Jn, define
ˆ
v(x, t)=vxˆn(x, t), tn
. (2.13)
Note that ˆv(x, tn−1,+)=vˆn−1,+(x)follows the characteristics forward from
tn−1totnto becomevn(x). If we use this type of test functions in (2.12), the last term in the right-hand side of (2.12) becomes zero. With this, summing the resulting equation overIi∈Ih, applying the boundary condition in (2.2), and assuming the continuity ofσ inI, we see that
φun, vn−φun−1,vˆn−1,++ i
σ ,∂vˆ ∂x
In i
−
j
Jnσ ν[v]ˆ |xˇn(xj,t)dt
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x
dt, v∈H1I
h
,
(2.14)
where with each characteristic a unit normalνis associated, the characteris-tic ˇxn(xj, t)emanates back from the boundary(cord, t∈Jn)ifj≥M+1 (seeFigure 2.3), andvis extended by zero outside ¯I. Note that we can formally write
i
σ ,∂vˆ ∂x
Iin=
Jn
σ ,∂vˆ ∂x
dt, (2.15)
andj
Jnσ ν[v]ˆ |xˇn(xj,t)dtrepresent the jumps across the characteristics. Now, invertain the second equation of (2.2), multiply byτ∈H1(I
h), inte-grate onIi∈Ih, and sum the resulting equation attn over allIi∈Ihto see that
M
i=1
a−1nσn−∂un
∂x , τ
Ii
Assuming thatu is continuous inI (so [u](xi)=0, xi∈Ᏽoh) and using the Dirichlet boundary condition in (2.2), (2.16) becomes
M
i=1
a−1nσn−∂un
∂x , τ
Ii
+
M
i=0
un{τν}| xi=
x∈ΓD
gnDτν|x, τ∈H1
Ih
.
(2.17)
From (2.14) and (2.17), we have the weak formulation on which the discontin-uous methods are based. Thus, we see that if(σ , u)is a solution of (2.2), then it satisfies (2.14) and (2.17); the converse also holds ifuis sufficiently smooth (e.g.,u∈H1(Ω)∩H2(I
h)). Namely, (2.14) and (2.17) are consistent with (2.2). The jump terms in the left-hand side of (2.14) are called theconsistent terms (which come from the Green formula), while the corresponding terms in (2.17) are termed the symmetric terms (which are added). Finally, notice that the stripsIni are oriented along the Lagrangian flow paths (i.e., the characteristic paths), and a fixed partition is utilized at the time level tn. These features suggest the name Eulerian-Lagrangian method [1,10].
3. The first mixed discontinuous method
3.1. The definition. LetVh×Whbe the finite element spaces for the approxi-mation ofσandu, respectively. They are finite dimensional and defined locally on each elementIi∈Ih, so letVh(Ii)=Vh|Ii andWh(Ii)=Wh|Ii,i=1, . . . , M. Neither continuity constraint nor boundary values are imposed onVh×Wh.
Based on (2.14) and (2.17), the first Eulerian-Lagrangian mixed discontinuous method for (2.1) is: find(σh, uh):{t1, . . . , tᏺ} →Vh×Whsuch that
φunh, v−φuhn−1,vˆn−1,++
M
i=1
σhn,∂v ∂x
Ii ∆tn−
M
i=0
σhnν[v]|xi∆t n
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x
dt, v∈Wh,
M
i=1
a−1nσn h−
∂unh
∂x , τ
Ii
+
M
i=0
un h
{τν}|xi=
x∈ΓD gn
Dτν|x, τ∈Vh, (3.1)
wherevis extended by zero outside ¯I. The initial approximation is given by
u0h=ᏼhu0, (3.2)
whereᏼhdenotes theL2-projection intoWh. That is,
We define the bilinear forms
Anτ1, τ2= M
i=1
a−1nτ1, τ2
Ii∆t
n, τ1, τ2∈V h,
Bn(τ, v)=
M
i=1
τ,∂v ∂x
Ii ∆tn−
M
i=0
{τν}[v]|xi∆t
n, τ∈V
h, v∈Wh. (3.4)
Then system (3.1) is of the form
φunh, v
−φunh−1,vˆn−1,+
+Bnσhn, v
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x
dt, v∈Wh,
Anσn h, τ
−Bnτ, un h
=
x∈ΓD gn
Dτν|x∆tn, τ∈Vh.
(3.5)
It can be seen [14] that the stiffness matrix arising from this system is positive definite for anyVhandWh. Also, this system is symmetric after changing a sign in either of the two equations. But this would alter the positive definiteness. As seen in the subsequent analysis, if (3.1) is written in a nonmixed form (or the standard Galerkin version), the positive definiteness and symmetry can be preserved simultaneously. Thus, in terms of implementation, it is desirable to write (3.1) in a nonmixed form. However, as pointed out in the introduction, we emphasize that the mixed formulation naturally stabilizes the discontinuous finite element method, seeSection 3.4. That is why we start with the mixed discontinuous method. For the advantages of the present mixed discontinuous methods over the classical mixed finite element methods, see [13,14].
LetIi∈Ihbe any element. Takev=1 onIiand zero elsewhere in the first equation of (3.1) to see that
φunh,1I i−
φunh−1,1Iˇi(tn−1)−
x∈∂Ii
σhnνIix∆tn
=
Jn
(f ,1)ˇIi(t)−
x∈ΓN∩∂ˇIi(t) gN|x−
x∈ΓD∩∂ˇIi(t) gnDbν|x
dt.
(3.6)
Proposition3.1. System (3.1) has a unique solution under (2.3).
Proof. Since (3.1) is a finite system, it suffices to show the uniqueness of the solution. Settingf=gD=gN=u0=0, it follows from (3.1) that
φuhn, v−φuhn−1,vˆn−1,+
+
M
i=1
σn h,
∂v ∂x
Ii∆
tn− M
i=0
σn hν
[v]|xi∆tn=0, v∈Wh,
M
i=1
a−1nσn h−
∂unh
∂x , τ
Ii∆
tn+ M
i=0
un h
{τν}|xi∆tn=0, τ∈Vh.
(3.7)
We complete the proof by an induction argument. Note thatu0h=0 by assump-tion and by (3.2). Letunh−1=0. The addition of the two equations in (3.7) with
v=un
handτ=σhnyields
φunh, unh
+a−1nσhn, σhn
∆tn=0, (3.8)
which implies, by (2.3), thatσhn=unh=0. Thus, the desired result follows.
3.3. Convergence. From now on, let
Vh
Ii
=Wh
Ii
=Ppi
Ii
, Ii∈Ih, pi≥0, (3.9)
wherePpi(Ii)is the set of polynomials of degree at mostpionIi. Note thatpi can be different on different elements.
Forv∈Hl(I
i), l≥0, there exists vh∈Ppi(Ii) and a positive constantC dependent onlbut independent ofv,h, andpisuch that [3]
v−vhHr(I i)≤
C(l)hmin(l,pi)+1−r i
max1, pi
l+1−r vHl+1(Ii), r=0,1,
v−vh(x)≤
C(l)hmin(l,pi)+1/2 i
max1, pi
l+1/2 vHl+1(Ii), x∈∂Ii.
(3.10)
We now state error estimates which will be shown inSection 6. Set
p=maxpi, Ii∈Ih, i=1, . . . , M
Forv∈H1(I×J), we define
∂ ∂zv
ˇ
xn(x, t), t=
∂ ∂tv
ˇ
xn(x, t), t+ϕ
∂ ∂xv
ˇ
xn(x, t), t, t∈Jn. (3.12)
Theorem3.2. Under (2.3), if(σ , u)and(σh, uh)are the respective solutions
to (2.2) and (3.1), then, for∆tsufficiently small,
max
1≤n≤ᏺu n−un
h
2
L2(I)+ ᏺ
n=1
σn−σn h
2
L2(I)∆tn
≤C(l)
M
i=1
h2 min(l,p) max(1, p)2l−3
∂u∂t
2
L2(J;Hl(I i))
+u2
L2(J;Hl+1(Ii))+u
2
L∞(J;Hl(Ii))
+∂u ∂z
2L2(J;Hl(I
i))
+σ2L2(J;Hl+1(Ii))
+ᏺ
n=1
∂σ∂x
2
L2(Ii×Jn)+ ∂2σ
∂z∂x
2L2(I
i×Jn)
+∂σ ∂t
2L2(I
i×Jn)
(∆t)2
, l≥0.
(3.13)
Note that the estimate inTheorem 3.2gives a suboptimal order inhof con-vergence, but it is sharp for (3.1) in the general case, as indicated by numerical experiments. In the case wherepifor allIi∈Ihis even, we can prove an optimal order inhof convergence for (3.1).
Theorem3.3. Under (2.3), if(σ , u)and(σh, uh)are the respective solutions
to (2.2) and (3.1) and ifpiis even for allIi∈Ih, then, for∆tsufficiently small,
max
1≤n≤ᏺu n−un
h
2
L2(I)+ ᏺ
n=1
σn−σn h
2
L2(I)∆tn
≤C(l)
M
i=1
h2 min(l,p)+2
max(1, p)2l−1
∂u∂t
2
L2(J;Hl+1(Ii))
+u2
L2(J;Hl+2(Ii))+u 2
L∞(J;Hl+1(Ii))
+∂u∂z 2
L2(J;Hl+1(Ii))
+σ2
L2(J;Hl+2(Ii))
+
ᏺ
n=1
∂σ∂x
2
L2(Ii×Jn)+
∂z∂x∂2σ 2L2(I i×Jn)
+∂σ ∂t
2L2(I
i×Jn)
(∆t)2
, l≥0.
(3.14)
3.4. Implementation. While method (3.1) is in mixed form, it can be im-plemented (if desired) in a nonmixed form [14]. We introduce the coefficient-dependent projectionsPn
h:L2(I)→Vhby
a−1nw−Pn hw
, τ=0 ∀τ∈Vh, (3.15)
forw∈L2(I), andRn
h:H1(Ih)→Vhby
M
i=1
a−1nRhn(v), τ
Ii= − M
i=0
[v]{τν}|xi+
x∈ΓD
gDnτν|x, τ∈Vh, (3.16)
forv∈H1(I
h). Using (3.15) and (3.16), (3.1) can be rewritten as follows [14]: finduh:{t1, . . . , tᏺ} →Whsuch that
φun h, v
−φun−1
h ,vˆn−1,+
+
M
i=1
Pn h
an∂u n h ∂x ,∂v ∂x Ii ∆tn
−
M
i=0
unhPhn
an∂v ∂x
ν
xi ∆tn
−
M
i=0
Pn h
an∂u n h
∂x
+Rn h un h ν
[v]|xi∆tn
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x dt
−
x∈ΓD gnDPhn
an∂v ∂x
ν|x∆tn ∀v∈Wh
(3.17)
withσhgiven by
σn h=Phn
an∂u n h
∂x
+Rn h un h . (3.18)
To see the relationship of (3.17) with traditional discontinuous Galerkin fi-nite element methods, we consider the case whereais piecewise constant. In this case, (3.17) becomes: finduh:{t1, . . . , tᏺ} →Whsatisfying
φunh, v−φuhn−1,vˆn−1,++
M
i=1
an∂u n h ∂x , ∂v ∂x Ii ∆tn
−
M
i=0
unh
an∂v ∂xν
x
i ∆tn
−
M
i=0
an∂u n h
∂x ν
[v]|xi∆t n+
M
i=1
a−1nRn h
un h
, Rn h(v)
Ii∆t
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x dt
−
x∈ΓD gDn
an∂v ∂x−R
n h
unh
ν|x∆tn ∀v∈Wh.
(3.19)
Note that without the terms involvingRh, (3.19) (as applied to elliptic prob-lems) is just the global element method introduced in [21]. With a positive sign in front of the fourth term in the left-hand side of (3.19) (and without theRh terms), it is the method recently introduced in [32]. TheRhterm in the left-hand sides of (3.17) and (3.19) naturally comes from the mixed formulation, which stabilizes the classical method and leads to higher convergence rates (in space). For more information on the relationship between the present MDFE method and other earlier methods, we refer to [13,14].
AlthoughRhappears, (3.19) can be evaluated virtually in almost the same amount of work as in the evaluation of the global element method. This is due to the definition ofRh in (3.16), where the matrix associated with the left-hand side can be diagonal if the basis functions ofVh are appropriately chosen. Also, (3.16) is totally local. The stiffness matrices arising from (3.17) and (3.19) are symmetric and positive definite. Numerical experiments will be given inSection 8.
4. The second mixed discontinuous method
4.1. The definition. Forv∈H1(I
h), define the one-sided limits at the nodes
xi∈Ᏽoh:
vxi+= lim
x→xi+v(x), v
xi−= lim
x→x−i v(x). (4.1)
At the endpointc, forτ∈Vh, we use the conventionτ(c+)=τ(c−); a similar meaning can be given atx=d. The second Eulerian-Lagrangian mixed discon-tinuous method for (2.1) is: find(σh, uh):{t1, . . . , tᏺ} →Vh×Whsuch that
φun h, v
−φun−1
h ,vˆn−1,+
+
M
i=1
σn h,
∂v ∂x
Ii∆
tn− M
i=0 σn
h
x+iν[v]|xi∆tn
=
Jn
(f ,v)ˆ − x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x
dt, v∈Wh,
M
i=1
a−1nσn h−
∂un h
∂x , τ
Ii
+
M
i=0
τxi+un h
ν|xi=
x∈ΓD gn
The second method differs from the first one in that the averaged quantities in (3.1) are replaced by the right-hand side limits. This method is an exten-sion of the local discontinuous Galerkin method proposed in [9] for advection-diffusion problems.
4.2. Existence and uniqueness. Existence and uniqueness of the solution to (4.2) can be shown as for (3.1).
Proposition4.1. Under (2.3), (4.2) has a unique solution.
4.3. Convergence. The finite element spacesVhand Whare defined as in (3.9). The convergence result inTheorem 3.2holds for (4.2) as well. As shown in [12] for elliptic problems, the optimality of convergence inhandpdepends on the type of boundary conditions. We consider the case wherec∈ΓD and
d∈ΓN. In this case, we have the following optimal convergence result.
Theorem4.2. Under (2.3), if(σ , u)and(σh, uh)are the respective solutions to (2.2) and (4.2),c∈ΓD, andd∈ΓN, then, for∆tsufficiently small,
max
1≤n≤ᏺu n−un
h
2
L2(I)+ ᏺ
n=1
σn−σn h
2
L2(I)∆tn
≤C(l)
M
i=1
h2 min(l,p)+2
max(1, p)2l+2
∂u∂t
2
L2(J;Hl+1(Ii))
+uL∞(J;Hl+1(Ii))
+∂u∂z 2
L2(J;Hl+1(Ii))
+σ2
L2(J;Hl+1(Ii))
+
ᏺ
n=1
∂σ∂x
2
L2(Ii×Jn)
+ ∂2σ ∂z∂x
2L2(I
i×Jn)
+∂σ∂t 2
L2(I i×Jn)
(∆t)2
, l≥0.
(4.3)
The proof of this theorem will be carried out inSection 6. The error esti-mate inTheorem 4.2is optimal in bothhandp. For other cases of boundary conditions, the error estimate inTheorem 3.2is sharp for (4.2). However, with an addition of appropriate boundary terms atx=c, dto (4.2), the resulting scheme can generate optimal error estimates. We will not pursue this; for more information in this direction, we refer to [12] for the treatment of elliptic prob-lems.
4.4. Implementation. System (4.2) can be implemented (if desired) in the same fashion as in (3.1). The operatorPn
h :L2(I)→Vhis defined as in (3.15), whileRn
h:H1(Ih)→Vhis modified by
M
i=1
a−1nRn h(v), τ
Ii= −
M
i=0 [v]xi
τx+i+
x∈ΓD gn
forv∈H1(I
h). Then (4.2) can be reduced to: finduh:{t1, . . . , tᏺ} →Whsuch that
φunh, v−φuhn−1,vˆn−1,++
M
i=1
Phn
an∂u n h ∂x ,∂v ∂x Ii ∆tn
−
M
i=0
unh
ν|xiPhn
an∂v ∂x
x+i
∆tn
−
M
i=0
Phn
an∂u n h
∂x
+Rhnunh
x+i[v]ν|xi∆t n
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x dt
−
x∈ΓD gn
DPhn
an∂v
∂x
ν|x∆tn ∀v∈Wh,
(4.5)
withσhgiven as in (3.18). Again, in the case in whichais piecewise constant, (4.5) becomes: finduh:{t1, . . . , tᏺ} →Whsatisfying
φuhn, v−φuhn−1,vˆn−1,+
+
M
i=1
an∂u n h ∂x , ∂v ∂x Ii ∆tn−
M
i=0
un h
ν|xian
∂v ∂x
xi+
−
M
i=0 an∂u
n h
∂x
xi+[v]ν|xi∆tn+ M
i=1
a−1nRn h
un h
, Rn h(v)
Ii∆t
n
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x dt
−
x∈ΓD gn
D
an∂v
∂x−R
n h un h
ν∆tn|
x ∀v∈Wh.
(4.6)
The stiffness matrices arising from (4.5) and (4.6) are symmetric and positive definite. Numerical results for (4.2) will be given inSection 8as well.
5. The third mixed discontinuous method
(σh, uh):{t1, . . . , tᏺ} →Vh×Whsuch that
φuhn, v−φuhn−1,vˆn−1,+
+
M
i=1
σn h,
∂v ∂x
Ii∆
tn− M
i=0 σn
h
x−iν[v]|xi∆tn
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x
dt, v∈Wh,
M
i=1
a−1nσn h−
∂un h
∂x , τ
Ii
+
M
i=0
τxi−un h
ν|xi=
x∈ΓD gn
Dτν|x, τ∈Vh. (5.1)
5.2. Existence and uniqueness. The analysis for (5.1) can be done in the same manner as for (4.2). We just state the corresponding results.
Proposition5.1. Under (2.3), (5.1) has a unique solution.
5.3. Convergence. The finite element spacesVhand Whare defined as in (3.9). Again, the convergence result inTheorem 3.2holds for (5.1). Similar to the second method, the optimality of convergence in h and p depends on the type of boundary conditions. The case wherec∈ΓN and d∈ΓD has the following optimal convergence result.
Theorem5.2. Under (2.3), if(σ , u)and(σh, uh)are the respective solutions
to (2.2) and (5.1),c∈ΓN, andd∈ΓD, then, for∆tsufficiently small,
max
1≤n≤ᏺu n−un
h
2
L2(I)+ ᏺ
n=1
σn−σhn2L2(I)∆tn
≤C(l)
M
i=1
h2 min(l,p)+2
max(1, p)2l+2
∂u∂t
2
L2(J;Hl+1(Ii))
+uL∞(J;Hl+1(Ii))
+∂u∂z 2
L2(J;Hl+1(Ii))
+σ2
L2(J;Hl+1(Ii))
+ᏺ
n=1
∂σ
∂x
2L2(I
i×Jn)
+ ∂2σ ∂z∂x
2L2(I
i×Jn)
+∂σ ∂t
2L2(I
i×Jn)
(∆t)2
, l≥0.
(5.2)
6. Proof of convergence results
6.1. Proof ofTheorem 3.2. To proveTheorem 3.2, we need three lemmas whose proofs can be found in [1,18].
Lemma6.1. With definition (2.13), for eachn,
φvˆn−1,+,vˆn−1,+−(φv, v)≤K∆tn(φv, v) ∀v∈L2(I). (6.1)
Lemma6.2. Ifv1∈L1(J;L2(I))andv2∈L2(I), then for a.e.t∈Jn,
v1,v2ˆ =v1, v2ˇ n=v1, v2ˇ +v1, v2ˇ n
(6.2)
for somen(x, t)=1+
n(x, t)such that
n(x, t)≤K
tn−t. (6.3)
Moreover, for∆tn small enough, there are positive constantsC1andC2such that
C1v1L2(I)≤v1ˇ L2(I)≤C2v1L2(I),
C1v2L2(I)≤v2ˆ L2(I)≤C2v2L2(I).
(6.4)
Lemma6.3. Forv∈H1(I×J), define
∂ ∂zv
ˇ
xn(x, t), t
=∂t∂ vxˇn(x, t), t
+ϕ ∂
∂xv
ˇ
xn(x, t), t
, t∈Jn. (6.5)
Then, if∆tnis small enough,
vn−vˇ L2(I)≤C
∆tn1/2
∂v∂zL2(I×Jn),
v−vˆL2(I)≤C∆tn1/2
∂v∂zL2(I×Jn).
(6.6)
We define the average
¯
vn= 1 ∆tn
Note that (2.14) can be written as follows:
φun, vn−φun−1,vˆn−1,++
M
i=1
¯
σn,∂v ∂x
Ii ∆tn−
M
i=0
¯
σnν[v]|xi∆t n
=
Jn
(f ,v)ˆ −
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x dt+
M
i=1
¯
σn,∂v
∂x
Ii∆
tn
−
i
σ ,∂vˆ ∂x
Ini +
i
Jnσ ν[v]ˆ |xˇn(xi,t)dt
−
M
i=0
¯
σnν[v]| xi∆t
n, v∈H1I
h.
(6.8)
We are now in a position to proveTheorem 3.2. Recall thatᏼh:L2(I)→W
h is the standardL2-projection. Also, letΠ
h:L2(I)→Vh be the standardL2 -projection. Below,is a positive constant independent ofhandp, as small as we please.
Proof ofTheorem3.2. Setηn=σ¯n−σhn and ζn=un−unh. The error equations follow from the subtraction of (3.1) from (6.8) and (2.17):
φζn, vn−φζn−1,vˆn−1,++
M
i=1
ηn,∂v ∂x
Ii∆t n−
M
i=0
ηnν[v]|xi∆t n
=
M
i=1
¯
σn,∂v
∂x
Ii
∆tn− i
σ ,∂vˆ ∂x
Iin
+
i
Jnσ ν[v]ˆ |xˇn(xi,t)dt− M
i=0
¯
σnν[v]|xi∆tn, v∈Wh,
M
i=1
a−1nηn−∂ζn
∂x , τ
Ii ∆tn+
M
i=0
ζn{τν}| xi∆t
n
=
M
i=1
a−1n(σ¯n−σn), τIi∆tn, τ∈Vh.
(6.9)
Set
ηn1=Πhσ¯n−σhn, η n
2=Πhσ¯n−σ¯n,
ζn
1=ᏼhun−unh, ζ2n=ᏼhun−un.
(6.10)
Note that ηn =ηn
1−ηn2 and ζn =ζ1n−ζ2n. Takev =ζ1n and τ=ηn1 in the
equations to see that
φζ1n, ζ1n
−φζ1n−1,ζˆ
n−1,+
1
+a−1nηn1, ηn1
∆tn
=φζ2n, ζ1n
−φζ2n−1,ζˆ
n−1,+
1
+
M
i=1
ηn2, ∂ζ1n
∂x
Ii ∆tn
−
M
i=0
ηn
2ν
ζn
1xi∆t
n+a−1nηn
2, ηn1
∆tn
−
M
i=1
∂ζ2n ∂x , η
n
1
Ii ∆tn+
M
i=0
ζ2n
ηn1νxi∆t n
+
M
i=1
¯
σn,∂ζ
n
1 ∂x
Ii
∆tn−
i
σ ,∂ζ1ˆ ∂x
Iin + i
Jnσ ν ˆ
ζ1xˇn(x i,t)dt−
M
i=0
¯
σnνζn
1xi∆t n + M
i=1
a−1nσ¯n−σn, ηn1
Ii∆tn
≡ 9
i=1 En
i
(6.11)
with the obvious definition ofEn
i, i=1, . . . ,9. Each of the terms in (6.11) is estimated as follows.
First, applyLemma 6.1to have
φζn
1, ζ1n
−φζn−1 1 ,ζˆ
n−1,+
1 ≥1 2 φζn
1, ζ1n
−φζn−1 1 , ζ1n−1
−Cζn
1 2
L2(I)∆tn.
(6.12)
Second, by Lemmas6.1,6.2, and6.3, we see that
En1=
φζn2−ζ2n−1
, ζ1n
+φζ2n−1−φζˇ2n−1, ζ1n
−φζˇ2n−1, ζ1nn
≤C
ζ1n
2
L2(I)+ζ2n−1 2
L2(I)
∆tn+∂ζ2∂t 2
L2(I×Jn)+
∂ζ2∂z 2L2(I×Jn−1) .
(6.13)
Third, by the definition ofΠh, we find that
Fourth, by a trace theorem, we observe that
En
3≤ Cp4
h
M
i=0
ηn
2
xi2∆tn+ζ1n 2
L2(I)∆tn. (6.15)
Fifth, by (2.3), it is obvious that
En4≤Cηn2 2
L2(I)∆tn+ηn1 2
L2(I)∆tn. (6.16)
Sixth, using the Green formula, it follows that
En5+E6n=
M
i=1
ζ2n, ∂ηn1
∂x
Ii ∆tn−
M
i=0
ζ2nν
ηn1xi∆tn=E n
5+E
n
6. (6.17)
By the definition ofᏼh, we get
En5=0, (6.18)
and, by a trace theorem,
En6≤ Cp4
h
M
i=0
ζn
2
xi2∆tn+ηn1 2
L2(I)∆tn. (6.19)
Seventh, the Green formula yields that
En7+E8n=
i ∂σ
∂x,ζ1ˆ
Iin− M
i=1
∂σ¯n
∂x , ζ
n
1
Ii
∆tn; (6.20)
consequently, by Lemmas6.2and6.3, we obtain
En
7+En8≤C
ζn
1 2
L2(I)+
∂σ∂x
2
L2(I×Jn)+ ∂σ∂z2L2(I
×Jn)
∆tn
∆tn. (6.21)
Eighth, with (2.3), we see that
En
9≤C
∂σ∂t2L2(I
×Jn)
∆tn2 +ηn
1 2
L2(I)∆tn. (6.22)
6.2. Proof ofTheorem 3.3. To showTheorem 3.3, we need the next lemma [20].
Lemma6.4. Withζ2n=ᏼhun−un andηn2=Πhσ¯n−σ¯n, if allpi are even,
i=1, . . . , M,
ζn
2
xi≤C(l) h
min(l,p)+3/2
max(1, p)l+3/2u
n
Hl+2(Ii∪Ii+1), l≥0, ηn
2
xi≤C(l)
hmin(l,p)+3/2
max(1, p)l+3/2σ¯
n
Hl+2(Ii∪Ii+1), l≥0.
(6.23)
With Lemma 6.4, Theorem 3.3 can be proven in the same fashion as for Theorem 3.2; to (6.15) and (6.19), we applyLemma 6.4instead of (3.10).
6.3. Proof ofTheorem 4.2. In the proofs of Theorems3.2and3.3, we have utilized the standardL2-projectionsᏼh:L2(I)→W
handΠh:L2(I)→Vh. To proveTheorem 4.2, we use a different pair of operators. Forχ∈H1(I
h), on each intervalIi=(xi−1, xi),ᏼhχ∈Whis determined by
ᏼhχ−χ, vIi=0 ∀v∈Ppi−1
Ii
ifpi≥1, ᏼhχxi−=χxi−
, (6.24)
andΠhχ∈Vhis given by
Πhχ−χ, v
Ii=0 ∀v∈Ppi−1
Ii
ifpi≥1,
Πhχ
xi+−1=χxi+−1. (6.25)
With this pair of operators, we have the next lemma.
Lemma 6.5. Letχ∈Hl+1(Ii)with Ii=(xi−1, xi)and l≥0. There exits a constantC(l)dependent only onlsuch that
ᏼhχ−χL2(I i)≤
C(l)hmin(l,pi)+1 i
max1, pi
l+1 χHl+1(Ii),
ᏼhχ−χ(x)≤C(l)h
min(l,pi)+1/2 i
max1, pi
l+1/2 χHl+1(Ii), x=xi−1, xi.
(6.26)
The same estimates hold forΠhχ−χ.
For simplicity, let the initial approximationu0
Proof oftheorem4.2. As for (6.11), it follows from (2.17), (6.8), and (4.2) that
φζn
1, ζ1n
−φζn−1 1 ,ζˆ
n−1,+
1
+a−1nηn
1, ηn1
∆tn
=φζn
2, ζ1n
−φζn−1 2 ,ζˆ
n−1,+
1
+
M
i=1
ηn
2, ∂ζ1n
∂x
Ii ∆tn
−
M−1
i=0 ηn2
xi+νζ1nxi∆t
n+a−1nηn
2, ηn1
∆tn
+
M
i=1
ζ2n, ∂ηn
1 ∂x
Ii ∆tn−
M
i=1 ζ2n
xi−
νηn1xi∆tn
+
M
i=1
¯
σn,∂ζ1n ∂x
Ii
∆tn− i
σ ,∂ζ1ˆ ∂x
Iin + i
Jnσ ν ˆ
ζ1xˇn(xi,t)dt− M
i=0
¯
σnνζ1nxi∆tn
+
M
i=1
a−1nσ¯n−σn, ηn
1
Ii∆t
n
≡ 9
i=1 Kin.
(6.27)
Now, by the definition ofΠhandᏼh, we see that
Kn
2 =Kn3=K5n=K6n=0. (6.28)
All other terms can be bounded as in the proof ofTheorem 3.2.
6.4. Proof ofTheorem 5.2. For the proof ofTheorem 5.2, the operatorsᏼh:
H1(I
h)→Whand Πh:H1(Ih)→Vhare accordingly modified as follows. For
χ∈H1(I
h), on each intervalIi=(xi−1, xi),ᏼhχ∈Whis determined by
ᏼhχ−χ, vI
i=0 ∀v∈Ppi−1
Ii
ifpi≥1, ᏼhχxi+−1
=χxi+−1
, (6.29)
andΠhχ∈Vhis given by
Πhχ−χ, vIi=0 ∀v∈Ppi−1
Iiifpi≥1,
Πhχ
x+i=χxi+. (6.30)
7. Generalizations
7.1. The case ofa≥0. So far, we have assumed that (2.3) holds. Whenais nonnegative only, letκ≥0 satisfy
a=κ2. (7.1)
Now, (2.2) is written in the form
∂(φu)
∂t +
∂
∂x(bu−κσ )=f inI×J,
σ=κ∂u
∂x inI×J,
u=gD onΓD×J,
(bu−κσ )νI=gN onΓN×J,
u(x,0)=u0(x) inI.
(7.2)
The corresponding weak formulation is defined as follows:
φun, vn−φun−1,vˆn−1,+
+
i
κσ ,∂vˆ ∂x
Ini −
i
Jnκσ ν[v]ˆ |xˇn(xi,t)dt
=
Jn
f ,vˆ− x∈ΓN
gNvˆ|x−
x∈ΓD
gDbνvˆ|x
dt, v∈H1Ih,
M
i=1
σn−κn∂u
n
∂x , τ
Ii+ M
i=0
unκnτνxi
=
x∈ΓD
gnDκnτν|x, τ∈H1
Ih
.
(7.3)
With this, the three families of Eulerian-Lagrangian mixed discontinuous meth-ods in the earlier sections can be accordingly defined. Moreover, all the stability and convergence results hold by combining the present techniques and those in [13, 14] for treating the case ofa≥0. Note that (7.3) applies to the case wherea≡0. In this case, only the inflow boundary condition is needed.
∂ ∂txˇn=ϕ
ˇ
xn, t
, t∈Jn,
ˇ
xn
x, tn=x.
(7.4)
In general, we cannot exactly follow the characteristic in (7.4), it can be fol-lowed only approximately. There are many ways to solve the first-order or-dinary differential equation (7.4). We consider only the Euler method [1]. For
tn−1,m=tn−1+m∆tn/Mn, withm=Mn, . . . ,1, set
ˇ
xn−1,m−1
n (x)=xˇn−1,m(x)−ϕ
ˇ
xn−1,m(x), tn−1,m∆tn
Mn, ˇ
xnn−1,Mn(x)=x.
(7.5)
Note that the number of stepsMncan depend onn. Denote by ˇx
n(x, t)the piecewise linear interpolant in time of ˇxn−1,m(x). If∆tn/Mn is sufficiently small (depending upon the smoothness ofϕ), the approximate characteristics do not cross each other, which is assumed here. Again, indicate the inverse of ˇ
xn(·, t)by ˆxn(·, t).
For anyt∈(tn−1,m−1, tn−1,m], define
˜
ϕ(x, t)=ϕxˇnxˆn(x, t), tn−1,m, tn−1,m, ˜b=ϕφ.˜ (7.6)
Then the approximate characteristics are defined in terms of ˜ϕ:
∂
∂txˇn=ϕ˜
ˇ
xn, t
, t∈tn−1, tn,
ˇ
xnx, tn=x.
(7.7)
Now, replacingbby ˜b+(b−b)˜ in the first equation of (2.2), (2.14) is modified as follows:
φnun, vn−φn−1un−1,vˆn−1,++
i
σ ,∂vˆ ∂x
Ini −
i
Jnσ ν
ˆ
vxˇn(x i,t)dt
=
Jn
f ,vˆ− x∈ΓN
gNvˆ|x−
x∈ΓD
gD˜bνvˆ|x dt
+
Jn ∂
∂x
(b˜−b)u,vˆ
−
x∈ΓN ˜
b−buνvˆ|x
dt, v∈H1Ih
,
(7.8)
Table8.1. The estimates foruh.
1/h 12 24 48 96
p error order error order error order error order
1 2.0489e-02 — 5.1661e-03 1.99 1.2923e-03 2.00 3.2276e-04 2.00 2 1.2717e-03 — 1.1590e-04 3.00 1.9853e-05 3.00 2.4817e-06 3.00 3 5.8455e-05 — 3.6763e-06 3.99 2.3015e-07 4.00 1.4401e-08 4.00 4 2.2652e-06 — 7.1221e-08 4.99 2.2320e-09 5.00 6.8973e-11 5.01
such that
φnun h, v
−φn−1un−1
h ,vˆn−1,+
+
M
i=1
σn h,
∂v ∂x
Ii ∆tn−
M
i=0
σn hν
[v]|xi∆tn
=
Jn
f ,vˆ−
x∈ΓN
gNvˆ|x−
x∈ΓD
gDbν˜ vˆ|x
dt, v∈Wh,
M
i=1
a−1nσhn−
∂unh ∂x , τ
Ii
+
M
i=0
unh
{τν}|xi=
x∈ΓD
gnDτν|x, τ∈Vh.
(7.9)
The second and third methods can be modified in the same manner. The ex-istence and uniqueness result previously obtained holds for (7.9) as well, and the convergence result holds provided the following assumption is satisfied:
φ, ϕ∈L∞J;W1,∞(I). (7.10)
8. Numerical results. We have observed numerical results for the three pre-sented methods similar to those obtained for the corresponding stationary problem in [12] when∆t is sufficiently small. As an example, we just report numerical results for the second mixed discontinuous method withc∈ΓDand
d∈ΓN. More numerical results can be found in [12]. The coefficients in (2.1) are constant (φ=1), the domain is the unit interval(0,1), and the exact solution is chosen as follows:
u(x, t)=exp(−at)1−cos2π (x−bt). (8.1)
considered uniform grids; analogous results have been observed on nonuni-form grids. These results are in a good agreement withTheorem 4.2. That is, the estimates are optimal in bothhandp. Also, similar numerical results are obtained forσh.
Acknowledgment. This work is supported, in part, by National Science Foundation Grants DMS-9626179, DMS-9972147, and INT-9901498, and by a gift grant from Mobil Technology Company.
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