discrete maximum principle for interior penalty discontinuous galerkin methods

Central European Journal of MathematicsDiscrete maximum principle for interior penalty discontinuous Galerkin methods Research Article Tamás L.. sétány 1/C, Budapest, 1117, Hungary Recei

Central European Journal of Mathematics

Discrete maximum principle for interior penalty

discontinuous Galerkin methods

Research Article

Tamás L Horváth1,2∗, Miklós E Mincsovics1,2†

1 Department of Applied Analysis and Computational Mathematics, Eötvös Loránd University, Pázmány P sétány 1/C,

Budapest, 1117, Hungary

2 MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Pázmány P sétány 1/C, Budapest, 1117, Hungary

Received 27 February 2012; accepted 29 April 2012

Abstract: A class of linear elliptic operators has an important qualitative property, the so-called maximum principle In this

paper we investigate how this property can be preserved on the discrete level when an interior penalty discon-tinuous Galerkin method is applied for the discretization of a 1D elliptic operator We give mesh conditions for the symmetric and for the incomplete method that establish some connection between the mesh size and the penalty parameter We then investigate the sharpness of these conditions The theoretical results are illustrated with numerical examples.

MSC: 65N30, 35B50

Keywords: Discrete maximum principle • Discontinuous Galerkin • Interior penalty

© Versita Sp z o.o.

1 Introduction

When choosing a numerical method to approximate the solution of a continuous mathematical problem we need to

consider which method results in an approximation that is not only close to the solution of the original problem, but

also shares the important qualitative properties of the original problem For linear elliptic problems the most important

qualitative property is the maximum principle The reader can find detailed explanations from different viewpoints about

the importance of the preservation of the maximum principle in [9, Sections 1,2], [13, Section 1], and [5]

∗ E-mail: thorvath@cs.elte.hu

† E-mail: mincso@cs.elte.hu

The preservation of the maximum principle was extensively investigated for finite difference methods (FDM) and for

finite element methods (FEM) with linear and continuous elements, but not in the context of the discontinuous Galerkin

method In this paper we take the first step to fill this gap Namely, we investigate an interior penalty discontinuous

Galerkin method (IPDG) applied to a 1D elliptic operator (containing diffusion and reaction terms) and we show that it

is possible to give reasonable and sufficient conditions for the maximum principle on the discrete level

The paper is organized as follows In Sections2and3we give a short overview on continuous and discrete maximum

principles including the important notions and preliminary results that we will use later on In Section4we deal with the

IPDG method applied to some 1D elliptic operator In Section5we give conditions under which the discrete maximum

principle holds In subsection 6.1we investigate the sharpness of our conditions with the help of numerical examples

We conclude the investigation in subsection6.2 We include an appendix about the Z- and M-matrices for the readers’

convenience since we use these notions throughout Section5

2 Continuous maximum principle for elliptic operators

We define the maximum principle for operators, following the book [8], instead of defining it for equations Naturally,

there are no important differences between the two approaches, but our choice is easier to handle Let Ω ⊂ R

d

be

an open and bounded domain with boundary ∂Ω, and Ω = Ω ∪ ∂Ω its closure We investigate the elliptic operator K ,

dom K = H1(Ω), defined in divergence form as

K u = −

d

X

i,j=1

∂

∂xj



aij ∂u

∂xi



+

d

X

i=1

bi ∂u

where a ij (x) ∈ C1

(Ω), bi (x), c(x) ∈ C (Ω). Note that smoothness of the coefficient functions gives the opportunity to

rewrite (1) to a non-divergence form that is more suitable for the investigation of maximum principles

Definition 2.1.

We say that the operator K , defined in (1), possesses the (continuous) maximum principle if for all u ∈ C

2

(Ω) ∩ C (Ω)

the following implication holds

Ω

u ≤maxn0, max

∂Ω uo.

Theorem 2.2 ([ 8 , Chapter 6.4, Theorem 2]).

If operator K , defined in(1), is uniformly elliptic and c ≥ 0, then it has the continuous maximum principle.

3 Maximum principle for FEM elliptic operators – short overview

3.1 The construction of the FEM elliptic operator

When discretizing the operator (1) with some finite element method we have to define the corresponding bilinear form

a (u, v ) =

Z

Ω

d

X

i,j=1

aij ∂u

∂xi

∂v

∂xj +

d

X

i=1

bi ∂u

where u ∈ H1(Ω), v ∈ H1(Ω) We note that this means we deal with nonhomogeneous Dirichlet boundary condition, for

the homogeneous one see Remark3.4

The following step is to define a mesh on Ω A 1D mesh consists of intervals The discrete maximum principle literature

focuses on regular triangle or hybrid meshes (containing both triangles and rectangles) in 2D and tetrahedron or block

meshes in 3D A given mesh determines the sets P = {x1, x2, , xN}and P∂ = {x N+1, xN+2, , xN +N ∂ }containing the vertices in Ω and on ∂Ω, respectively. Let us introduce two more notations: N = N + N ∂ and P = P ∪ P ∂.

Next we can define a subspace of H1(Ω) corresponding to the mesh This can be done by giving a basis of this subspace The basis functions are denoted by Φi (x), i = 1, , N The discrete maximum principle literature investigates almost

solely the case of hat-functions which are defined with the following properties:

1 the basis functions are continuous;

2 the basis functions are piecewise linear over triangles/tetrahedrons and multilinear over rectangles/blocks;

3 Φi(xi ) = 1 for i = 1, , N ;

4 Φi(xj ) = 0 for i, j = 1, , N , i 6= j

Note that due to such choice,

1 the subspace consists of continuous functions;

2

PN

i=1φi (x) = 1 holds for all x ∈ Ω;

3 Φi (x) ≥ 0 holds for all x ∈ Ω and i = 1, , N ;

4 in a linear combination of the basis functions the coefficients represent the values of the resulting function at the

points of P

(We remark that for higher order elements the investigation is more difficult, and positive results are obtained only for

a simple 1D problem, see [17]; for a higher dimensional case, in [12] negative results are obtained.)

Finally, we can construct the so-called stiffness matrix K ∈ R

N×N

as

Kij = a(Φ j,Φi ),

that is, the discrete operator corresponding to (1 In the following it will be useful to introduce the partitioned form

K = [K0|K∂], where K0 ∈ R N×N

, K∂ ∈ R N×N ∂

, acting on the vector u = [u0|u∂]T

∈ R N

, u0 ∈ R N

, u∂ ∈ R N ∂

, which is

constructed by taking into consideration the separation of the (discrete) interior and boundary nodes

3.2 Maximum principle for FEM elliptic operators

To define the corresponding discrete maximum principle we introduce some notation The symbol 0 denotes the zero

matrix (or vector), e is the vector all coordinates of which are equal to 1. The dimensions of these vectors and matrices

should be clear from the context Inequalities A ≥ 0 or a ≥ 0 mean that all elements of A or a are nonnegative By max a

we denote the maximal coordinate of the vector a. Now we are ready to define the corresponding discrete maximum

principle for the matrix K.

Definition 3.1 ([ 4 ]).

We say that a matrix K has the discrete maximum principle if the following implication holds:

Ku ≤ 0 =⇒ max u ≤ max {0, max u ∂}.

Note that this definition is adequate only because the chosen basis functions have special properties E.g., for higher

order basis functions this definition is not applicable It is relatively easy to give sufficient and necessary conditions for

this principle

Theorem 3.2 ([ 4 ]).

The matrix K possesses the discrete maximum principle if and only if the following three conditions hold:

(T1) K−1≥ 0,

(T2) −K −1K∂ ≥ 0,

(T3) −K −1K∂ e ≤ e.

Theorem3.2is a theoretical result and difficult to apply directly Usually these conditions are relaxed with the following

practical conditions

Theorem 3.3 ([ 4 ]).

The matrix K has the discrete maximum principle if the following three conditions hold:

(P1) K0is a nonsingular M-matrix,

(P2) −K∂ ≥ 0,

(P3) Ke ≥ 0.

For the definition of M-matrix see Definition7.2.The reader can find detailed information and a plentiful reference list

about the discrete maximum principle in [16] For attempts to use less restrictive practical conditions we recommend the

papers [15] and [10]

Remark 3.4.

Note that if we apply the homogeneous Dirichlet boundary condition  this is the case when we eliminate the boundary

condition at the continuous level  then the matrix K∂has no effect, which results in that we need to guarantee (T1) or

(P1) only This milder property has its own name, the so-called nonnegativity preservation property

If we want to handle the homogeneous Dirichlet boundary condition abstractly, we have to introduce a new bilinear form

a0

, formally the same as (2) with the exception that it is defined for u ∈ H

1

(Ω), v ∈ H

1

(Ω) Then by the discretization

we simply do not have K∂.

4 Discontinuous Galerkin method  problem setting and discretization

Discontinuous Galerkin methods have been thoroughly investigated in recent years [1,2,11] These methods have several

advantages:

• built-in stability for time-dependent advection–convection equations,

• adaptivity can be easily done (the basis functions do not have to be continuous over the interfaces),

• the mesh does not have to be regular, hanging-nodes can be handled easily,

• conservation laws could be achieved by numerical solutions

For more details see, e.g., [6,7,14] The idea behind the discontinuous Galerkin method in comparison with FEM with

piecewise linear and continuous basis functions is to get better approximations and to spare computational time by

dropping the continuity requirement (even in the case when the solution of the original problem is continuous, as it

holds for many applications)

4.1 Problem setting

Let us set Ω = (0, 1) and consider the following special elliptic operator K , dom K = H

1

(0, 1), defined as

K u = −(pu 0

)

0 + k

where p, k ∈ R , p > 0. It is clear that for this operator the maximum principle holds due to Theorem 2.2 Here we

remark that the space H

1

(0, 1) consists of continuous functions.

Note that continuity is an important qualitative property and it cannot be preserved by the discontinuous Galerkin

method This is one of the reasons why we need to be careful, especially with the preservation of some milder qualitative

properties which are in connection with the continuity This leads directly to the investigation of maximum principle for

the discontinuous Galerkin method There are several sorts of discontinuous Galerkin methods in the literature In this

paper we will consider the interior penalty discontinuous Galerkin method

4.2 The construction of the IPDG elliptic operator

As opposed to the standard FEM approach, here the first step to discretize the operator (3) with the interior penalty

discontinuous Galerkin method is to define a mesh on (0, 1) Let us denote it by τ h and define in the following way:

0 = x0< x1< x2< < xN−1< xN = 1. We use the notations I n = [x n−1, xn ], h n = |I n | , h n− 1,n = max {h n−1, hn}(with

h 0,1 = h1, h N,N+1= h N).

The next step is to define the space Dl (τ h) = {v : vI n ∈ Pl (I n ), n = 1, 2, , N } – piecewise polynomials over every interval with maximal degree l For these functions we introduce the right and left hand side limits v (x

+

n) =

limt→0+v (x n + t), v (x n −) = limt→0+v (x n − t), and jumps and averages over the mesh nodes as

[[u(x n )]] = u(x n − ) − u(x+

n ), {{u (x n )} }= 1

2

(u(x

−

n ) + u(x

+

n )).

At the boundary nodes these are defined as

[[u(x0)]] = −u(x

+

0), {{u (x0)} } = u(x+

0), [[u(x N )]] = u(x − ), {{u (x N )} } = u(x − ).

We fix the penalty parameter σ ≥ 0 and ε whichcan be any arbitrary number, but is usually chosen from the set {− 1, 0, 1}. The value ε = 1 gives the nonsymmetric, ε = 0 the incomplete, and ε = −1 the symmetric IPDG In [2]

several DG methods were examined, and conditions for the convergence were collected The nonsymmetric version

converges for all σ > 0, while the two other converge only for σ > σ ∗

, where σ

∗

is unknown for both methods The

symmetric method is the only one that guarantees optimal convergence order, because the symmetric version is the only

one that is adjoint consistent After these preparations we are ready to define the (discrete) IPDG bilinear form as

aDG(u, v ) =

N−1

X

n=0

x n

+1

Z

x n

pu 0 (x ) v 0 (x ) dx −

N

X

n=0 {{pu 0 (x n )} } [[v (x n )]] + ε

N

X

n=0 {{pv 0 (x n )} } [[u(x n)]]

+

N

X

n=0

σ hn,n+1[[v (x n )]] [[u(x n)]] +

Z 1

0

k2uv dx

Note that fixing the parameters σ , ε and the mesh τ hcan be done in parallel The crucial step is the following We fix a

basis in the space D l (τ h ) First we need to choose l = 1 for the same reasons as in the FEM case discussed in Section3.

When choosing the basis functions we need to consider the following If we want to use the Definition3.1and apply

Theorems3.2and3.3, then we need to choose basis functions with the important properties listed in subsection3.1 We

already set aside continuity, but the next choice fulfils the second and third property and a milder version of the fourth,

and this is enough for us

We will use Φ1i (x ) for the (2(i − 1) + 1)thbasis functions, and Φ2i (x ) for the (2(i − 1) + 2)thbasis functions, see Figure1.

On interval I ithe function Φ1

i (x ) is the linear function with Φ

1

i (x

+

i−1) = 1, Φ

1

i (x

−

i) = 0 and Φ

2

i (x ) is the linear function with

Φ

1

i (x

+

i−1) = 0, Φ

1

i (x

−

i ) = 1, and these functions are zero outside I i, see Figure1.

Figure 1. Φ1i (x ) and Φ2i (x )

Finally, we construct the IPDG elliptic operator similarly to the way we did in the previous section However there

are slight differences This matrix can be split in a partitioned form by separating the (discrete) interior and boundary

nodes as

K=

"

K0 K∂

A B

#

,

where K ∈ R

(2N )×(2N )

, K 0∈ R (2N −2)×(2N −2)

, and the others are trivial The 2Nbasis functions are ordered as follows:

the first 2N − 2 are the basis functions that belong to the interior nodes and they are numbered from left to right. The

(2N − 1)

th

belongs to the left boundary and the 2N

th

belongs to the right boundary Note that the matrices A and B

are not important from the point of view of the maximum principle, thus we can omit them So the matrix we need to

investigate has the usual form K = [K0|K∂].

Remark 4.1.

When working with the homogeneous Dirichlet boundary condition we could restrict aDG to D0× D0

, where D0(τ h) =

{v ∈ D1(τ h ) : v (0) = v (1) = 0} (Φ1

(x ) and Φ

2

N (x ) are excluded from the basis), although this is not a usual practice in

the discontinuous Galerkin community Let us denote the corresponding bilinear form by a0DG and define it as

a0

DG(u, v ) =

N−1

X

n=0

x n

+1

Z

x n

pu 0 (x ) v 0 (x ) dx −

N−1

X

n=1 {{pu 0 (x n )} } [[v (x n )]] + ε

N−1

X

n=1 {{pv 0 (x n )} } [[u(x n)]]

+

N−1

X

n=1

σ hn,n+1 [[v (x n )]] [[u(x n)]] +

Z 1

0

k2uv dx

In this case the discrete operator simplifies to K0 and, similarly to Remark3.4, only (T1) or (P1) should be fulfilled

In the following we calculate the elements of the matrix K.

4.3 The exact form of the discrete operators

It is easy to check that ∂ xΦ1

i (x ) = −1/h i , ∂ xΦ2

i (x ) = 1/h i, which means that the averages are



∂ xΦ

1

i (x k)

= −

1

2h i

∂ xΦ

2

i (x k)

= 1

2h i

at both endpoints x k of I i, with the exception of the boundary nodes, where there is no division by 2 Similarly, the

jumps are



Φ

1

i (x i−1)



= −1,



Φ

2

i (x i)



= 1

and zero elsewhere Using these facts we can calculate the matrix entries Summing them up we have the following

discretization matrices:

K0=























d1 r1 s2

t2 e2 q2w2

w2q2 d2 r2 s3 s

2 t

3 e

3 q

3 w

3

wi qi di ri si+1 si−1 ti ei qi wi

wN−1 qN−1 dN−1rN−1 sN−

1 tN eN





































v

1 0

s1 0

0 0

0 0

0 sN

0 vN















,

where

di= p 2h i +

σ hi,i+1 +

pε 2h i + k

2hi

3

, i = 1, , N − 1,

ei= p 2h i +

σ hi− 1,i+

pε 2h i + k

2hi

3

, i = 2, , N ,

wi= pε 2h i , i = 2, , N − 1,

qi = − p

hi +

p 2h i

− pε 2h i + k

2hi

6

, i = 2, , N − 1,

ri=

p 2h i+1

hi,i+1 − pε 2h i , i = 1, , N − 1,

si = − p 2h i , i = 1, , N ,

ti= p 2h i−1

hi− 1,i − pε 2h i , i = 2, , N ,

vi = − p

hi +

p 2h i

− pε

hi + k

2hi

6

, i = 1, , N ,

and zero elsewhere

5 Maximum principle for IPDG elliptic operators

Our aim is to get useful mesh conditions that guarantee the discrete maximum principle by using Theorem3.3 First we

deal with (P1) To this aim, we ask for the diagonal elements of the matrix K 0to be nonnegative and the off-diagonal elements to be nonpositive

• di, ei We get the following conditions for ε:

ε ≥ − 1 − 2σ h i

phi,i

+1

− 2k

2h2

i

3p , i = 1, , N − 1,

ε ≥ − 1 − 2σ h i

phi− 1,i −

2k2h2

i

3p , i = 2, , N

• wi wishould be nonpositive, which indicates

in the case where we have more than two subintervals See the third part of Remark5.4for the degenerate case

This means that ε = 1 is excluded generally.

• qi Because of q i we need to guarantee −p/(2h i ) − pε/(2h i ) + k2hi/ 6 ≤ 0, i = 2, , N − 1, which means the

following for ε:

ε ≥ −1 +k2h2i

3p , i = 2, , N − 1.

Or, rephrasing it for the mesh, we have h

2

i ≤ 3(1 + ε) p/k2

, i = 2, , N − 1, in the case where k 6= 0 (In the case

k = 0 we simply have ε ≥ −1.)

• si The inequality s i <0 always holds.

• ri , t i We need to guarantee p/(2h i+1) − σ /h i,i+1− pε/ (2h i ) ≤ 0 and p/(2h i−1) − σ /h i− 1,i − pε/ (2h i ) ≤ 0. After

re-indexing t i and reformulating we have

hi,i+1

hi+1 −

εhi,i+1

hi ≤ 2σ

hi,i+1

hi −

εhi,i+1

hi+1 ≤

2σ

p , i = 1, , N − 1. (5)

Finally, we show that there is no other restriction needed since the following lemma is valid

Lemma 5.1.

There exists a positive vector v with K0v > 0.

Proof. First let us consider the case where k = 0 and p = 1. We choose the dominant vector v as the piecewise

linear interpolation of the function d(x ) = c − x

2

with the bases of Φ

j

i in the interior nodes and zero at x = 0, 1, where

c ≥1, see Figure2 We prove that this choice is suitable

Let us denote this interpolation by Πd (x ) and let v contain its coefficients, so Π d (x ) =P

(i,j )∈int(τ h)v 2(i−1)+j −1Φj i (x ), where

the summation goes over all basis functions with exception of the two that belong to the boundary nodes (Φ

1

(x ) and

Φ

2

N (x )). It is clear thatv > 0 and we need to prove that K0v > 0 holds. The meaning of this inequality is that

aDG(Πd (x ), Φ

j

i (x )) > 0 holds for all basis functions, since, for example, for the first coordinate of K0v,

(K 0v)1 = X

(i,j )∈int(τ h)

v 2(i−1)+j −1 aDG(Φj i (x ), Φ2

(x )) = aDG

X

(i,j )∈int(τ h)

v 2(i−1)+j −1Φj i (x ), Φ2

(x )

!

= aDG(Πd (x ), Φ2

(x )).

Next we calculate these bilinear forms The function Πd (x ) is continuous, therefore its jumps are zero all over the nodes,

which means we have to take into account neither ε, nor the penalty terms.

Figure 2. Πd (x ) for c = 1.3

The derivative of Πd (x ) can be calculated on every I n. It is

c − x2

x1 on I1, − x

2

i − x2

i−1

xi − xi−1 = − x i − xi−1 on Ii, i = 2, , N − 1, x

2

N−1− c

1 − x N−1 on

IN

This means

aDG Πd (x ), Φ2

(x )

= Z

I

1

∂xΠd (x ) ∂ xΦ2

(x ) dx − { {∂xΠd (x1)} }

Φ

2

(x1)



=

 c − x2

x1

 Z

I

1

1

h1dx

| {z }

=1

−



(c − x

2

)/x1− x1− x2

2



·1 = c − x

2

2x1

+

x1+ x2

2

Similarly,

aDG Πd (x ), Φ1

(x )

=

c − x2

2x1

+

x1+ x2

2

.

For i 6= 1, N − 1, N ,

aDG Πd (x ), Φ2

i (x )

= Z

I i

∂xΠd (x ) ∂ xΦ2

i (x ) dx − { {∂xΠd (x i )} }

Φ

2

i (x i)



= − (x i + x i−1)

Z

I i

1

hi dx −



− xi + x i−1+ x i + x i+1

2



·1 = xi+1− xi−1

2

.

(7)

For i 6= 1, 2, N ,

a

DG Πd (x ), Φ

1

i (x )

= Z

I i

∂xΠd (x ) ∂ xΦ

i (x ) dx − { {∂xΠd (x i−1)} }

Φ

2

i (x i−1)



= − (x i + x i−1)

Z

I i

−1

hi dx −



− xi + x i−1+ x i−1+ x i−2

2



· (−1) = xi − xi−2

2

.

(8)

On I N−1,

aDG Πd (x ), Φ2

N−1(x )

= Z

I N−

1

∂xΠd (x ) ∂ xΦ2

N−1(x ) dx − { {∂xΠd (x N−1)} }

Φ

2

N−1(x N−1)



= − (x N−2+ x N−1)

Z

I N−

1

1

hN−1dx −

1

2



− (x N−2+ x N−1) +

x2

N−1− c

1 − x N−1



·1

= − xN−2+ x N−1

2 +

c − x2

N−1

2(1 − x N−1)

.

(9)

Finally,

aDG Πd (x ), Φ1

N (x )

= − xN−2+ x N−1

2 +

c − x2

N−1

2(1 − x N−1)

.

We have to prove that these are positive values The first three (6)–(8) are trivial To prove that (9) is positive, some

simple calculation is still needed

− xN−2+ x N−1

2 +

c − x2

N−1 2(1 − x N−1)

> 0, c − x

2

N−1

1 − x N−1

> xN−2+ x N−1,

and this holds since

c − x2

N−1

1 − x N−1 =

(

√

c − xN−1)(√ c + x N−1)

1 − x N−1

> √ c + x N−1> 1 + x N−1> xN−2+ x N−1.

When p 6= 1, we only have to multiply the matrix K0by p, which makes no difference in the sign of the product. When

k 6= 0, we have the extra terms R

I i k2

Φ

j

i (x ) · Φ l (k ), where j , l ∈ {1, 2}. All functions are positive, so these integrals are

also positive We have just increased the elements of K 0 , consequently increased the coordinates of K 0v.

Accordingly, we can apply Theorem7.3, which completes the investigation of the condition (P1) Property (P2) means

that v1 and v Nshould be nonpositive, i.e.,

ε ≥ − 3p + k

2h2

i

6p

= −

1

2 +

k2h2

i

6p

≥ −1 2

Note, it means that ε = −1 is excluded Property (P3) means that 0 ≤ (K 0|K∂) e should hold. It is equivalent to

aDG(1, Φ

j

i ) ≥ 0 for (i, j ) ∈ int (τ h), for example, for the first coordinate of (K 0|K∂) e:

((K 0|K∂) e)1 =

N

X

i=1

2

X

j=1

1 · aDG Φ

j

i (x ), Φ

2

(x )

= aDG

N

X

i=1

2

X

j=1

1 · Φ j

i (x ), Φ

2

(x )

!

= aDG(1, Φ

2

(x )).

The result of this matrix-vector product is

















k2h1

2

− ε p

h1

k2h2

2

k2hN−1

2

k2hN

− ε p hN















