Galerkin discretizations of 4th order in space and time ... · obtain such superconvergence results...

International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:04 1

131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S

Galerkin discretizations of 4th order in space and time for the

Convection-Diffusion-Reaction (CoDiRe) equation

R. Mahmood∗, S. Hussain†

June 15, 2013

Abstract

In this paper, we extend the continuous Galerkin-Petrove time discretization scheme

studied in [5], for the nonstationary Convection-Diffusion-Reaction (CoDiRe) equation.

In particular, we analyze the 4th order cGP(2)-method and compare it with existing low

order method. Moreover, for the approximation in space we use the nonparametric Q3-

element which belongs to a family of recently derived higher order nonconforming finite

element spaces and leads to an approximation error in space of order 4, too, in the L2-

norm. We also combine the space discretization with edge oriented jump stabilization

(EOJ) in order to get the stable discretization for the increasing convection. We discuss

implementation aspects of the time discretization as well as efficient multigrid methods

for solving the resulting block systems which lead to convergence rates being almost

independent of the mesh size and the time step. The expected optimal accuracy of the full

discretization error of 4th order in space and time is confirmed by several numerical tests.

In our numerical experiments we compare different spatial and temporal discretization

approaches with respect to accuracy and computational cost.

Keywords: continuous Galerkin-Petrov method, convection-diffusion equation, multigrid

method

2000 Mathematics Subject Classification (MSC): 65M12, 65M55, 65M60

1 Introduction

For solving nonstationary flow problems, it is very common to discretize partial differential

equations (PDEs) first in space and then in time, known as the ‘Method of Lines’. This

approach creates a system of ordinary differential equations (ODEs) which might be solved

∗Air University Islamabad, Pakistan†Mohammad Ali Jinnah University, Islamabad, Pakistan

1



2 R. Mahmood, S. Hussain

by state-of-the-art ODE integrators. However, the grid points of the spatial mesh have to stay

fixed in time or are often subject to certain constraints, as for example in the case of moving

mesh methods, so that these methods often have difficulties in changing the spatial mesh from

time step to time step. On the other hand, the Rothe method, which performs first the semi-

discretization in time, allows a fully adaptive integration of time dependent PDEs. A class

of time discretization schemes which is based on Rothe’s method is the continuous Galerkin-

Petrov discretization (cGP(k)-methods) and the discontinuous Galerkin (dG(k)) approach.

The cGP-method has already been used by Aziz and Monk [2] (but not under this name) for

the linear heat equation in which case they could prove optimal error estimates as well as

superconvergence results at the discrete time points τn. Currently, extensive tests regarding

the higher order accuracy in time have been performed for the heat equation in [5].

In this paper, we extend these numerical studies to the nonstationary Convection-Diffusion-

Reaction (CoDiRe) equation. In particular, we implement and analyze numerically the (fully

implicit) cGP(2)-method which is found, at comparable numerical cost per time step, to be of

higher order than classical schemes like Crank-Nicolson or BDF methods, namely of order 3

in the whole time interval and superconvergent of order 4 in the discrete time points. Since we

obtain such superconvergence results at tn. Moreover, the corresponding spatial discretiza-

tion is carried out by using biquadratic finite elements (Q2) in [5]. As corresponding 4th

order space discretization, we use the standard Galerkin Finite Element Method (FEM) with

higher order nonconforming (nonparametric) quadrilateral Q3-elements (see [6]). In order to

get the stable discretization, we have also combine the underlying spatial discretization with

the edge-oriented jump stabilization (EOJ) [12] for the convective term. We use such non-

conforming elements since they show an advantageous numerical behaviour for saddle-point

problems, particularly for incompressible flow problems together with discontinuous pressure

approximations, and they are preferable for parallel computing due to the fact that they only

require edge- or face-oriented communication which simplifies the parallel data exchange. To

solve the associated linear (block) systems, we propose a geometrical multigrid solver with

canonical grid transfer operators due to the FEM space Q3. The numerical experiments con-

firm that such multigrid methods [3, 6, 7, 11] are very efficient solvers since their rate of

convergence is almost independent of the space mesh size and the size of the time step on

structured as well as on semi-structured meshes.

We compare in numerical experiments our new space-time discretization of order 4 with

the standard Crank-Nicolson scheme of 2 concerning the achieved accuracy in relation to the

required CPU-time. The results clearly show the big advantage of the developed high order

methodology and the superior computational complexity during grid refinement.

To solve the associated linear block systems, we apply a geometrical (block) multigrid



Galerkin time discretizations of 4th order in space and time 3

solver. The numerical experiments confirm that multigrid methods [3, 11] can beregarded as

the most efficient iterative solvers since their rate of convergence is almost independent of the

problem size which is characterized here by the mesh size of the space grid and the size of

the time step.

2 The cGP-method for the CoDiRe equation

We consider the nonstationary Convection-Diffusion-Reaction equation: Find u : Ω× [0, T ] →R such that

dtu− αu+ β · ∇u+ γu = f in Ω× (0, T ),

u = 0 on ∂Ω× [0, T ],

u(x, 0) = u0(x) for x ∈ Ω,

(1)

where u(x, t) is the quantity of interest in the point x ∈ Ω at time t ∈ [0, T ], α denotes

the diffusion coefficient, β := (β1, β2) represents the convection velocity, γ is the reaction

coefficient, f : Ω × (0, T ) → R a given source term and u0 : Ω → R the initial temperature

field at time t = 0. For simplicity, we assume homogeneous Dirichlet conditions at the

boundary ∂Ω of a polygonal domain Ω ⊂ R2.

We start with the time discretization of problem (1) which is of variational type. In the

following, let I = [0, T ] be the time interval with some positive final time T . For a function

u : Ω× I → R and a fixed t ∈ I we will denote by u(t) := u(·, t) the associated space function

at time t which is an element of a suitable function space V . In case of the heat equation,

this space is the Sobolev space V = H10 (Ω). In order to characterize functions t 7→ u(t) we

define the space C(I, V ) as the space of continuous functions u : I → V equipped with the

norm

‖u‖C(I,V ) := supt∈I

‖u(t)‖V

and the space L2(I, V ) containing discontinuous functions as

L2(I, V ) := u : I → V : ‖u‖L2(I,V ) <∞ , ‖u‖L2(I,V ) :=

(∫

I‖u(t)‖2V dt

)1/2

.

In the time discretization, we decompose the time interval I into N subintervals In :=

[tn−1, tn], where n = 1, . . . , N and 0 = t0 < t1 < · · · < tN−1 < tN = T. The symbol τ

will denote the time discretization parameter and will also be used as the maximum time

step size τ := max1≤n≤N τn, where τn := tn − tn−1. Then, we approximate the solution

u : I → V by means of a function uτ : I → V which is piecewise polynomial of some order k

with respect to time, i.e., we are looking for uτ in the discrete time space

Xkτ:= u ∈ C(I, V ) : u

∣∣In

∈ Pk(In, V ) ∀ n = 1, . . . , N, (2)




where

Pk(In, V ) :=u : In → V : u(t) =

k∑

j=0

U jtj , ∀ t ∈ In, Uj ∈ V, ∀ j

.

We introduce the discrete time test space

Y kτ:= v ∈ L2(I, V ) : v

∣∣In

∈ Pk−1(In, V ) ∀ n = 1, . . . , N (3)

consisting of piecewise polynomials of order k− 1 which are globally discontinuous at the end

points of the time intervals. Now, we multiply the first equation in (1) with a test function

vτ ∈ Y kτ, integrate over Ω × I, use Fubini’s Theorem and partial space integration of the

Laplacian term and obtain the following time discrete problem: Find uτ ∈ Xkτ

such that

uτ(0) = u0 and

∫ T

0

(dtuτ(t), vτ(t))Ω + a(uτ(t), vτ(t))

dt =

∫ T

0(f(t), vτ(t))Ω dt ∀ vτ ∈ Y k

τ, (4)

where (·, ·)Ω denotes the usual inner product in L2(Ω) and a(·, ·) the bilinear form on V × V

defined as

a(u, v) :=

∫

Ωα(∇u · ∇v) + (β · ∇u, v) + γ(u, v) dx ∀ u, v ∈ V.

We will call this discretization the exact continuous Galerkin-Petrov method of order k or

briefly the ”exact cGP(k)-method”. The name Galerkin-Petrov is due to the fact that the

test space Y kτ

is different from the ansatz space Xkτ. With ”exact” we indicate that the time

integral at the right hand side is evaluated exactly.

Since the discrete test space Y kτ

is discontinuous, problem (4) can be solved in a time

marching process where successively local problems on the time intervals are solved. There-

fore, we choose test functions vτ(t) = vψn,i(t) with an arbitrary time independent v ∈ V and

a scalar function ψn,i : I → R which is zero on I \ In and a polynomial of order less or equal

k − 1 on In. Then, we obtain from (4) the ”In-problem”: Find uτ|In ∈ Pk(In, V ) such that

∫

In

(dtuτ(t), v)Ω + a(uτ(t), v)

ψn,i(t) dt =

∫

In

(f(t), v)Ω ψn,i(t) dt ∀ v ∈ V (5)

for i = 1, . . . , k, with the ”initial condition” uτ|In(tn−1) = uτ|In−1(tn−1) for n ≥ 2 or

uτ|In(tn−1) = u0 for n = 1.

To determine uτ|In we represent it by a polynomial ansatz

uτ(t) :=k∑

j=0

U jnφn,j(t) ∀ t ∈ In, (6)




where the ”coefficients” U jn are elements of the Hilbert space V and the real functions φn,j ∈

Pk(In) are the Lagrange basis functions with respect to k + 1 suitable nodal points tn,j ∈ In

satisfying the conditions

φn,j(tn,i) = δi,j, i, j = 0, . . . , k (7)

with the Kronecker symbol δi,j. In [10], the tn,j have been chosen as the quadrature points

of the (k + 1)-point Gauß-Lobatto formula on In. Here, we take another choice: For an easy

treatment of the initial condition for (5), we set tn,0 = tn−1. Then, the initial condition is

equivalent to the condition

U0n = uτ|In−1

(tn−1) if n ≥ 2 or U0n = u0 if n = 1. (8)

The other points tn,1, . . . , tn,k are chosen as the quadrature points of the k-point Gauß formula

on In. This formula is exact if the function to be integrated is a polynomial of degree less or

equal 2k − 1. From the representation (6) we get

∫

In

(dtuτ(t), v)Ω ψn,i(t)dt =k∑

j=0

(U jn, v)Ω

∫

In

φ′n,j(t)ψn,i(t) dt ∀ v ∈ V. (9)

We define the basis functions φn,j ∈ Pk(In) of (6) via the affine reference transformation

Tn : I → In where I := [−1, 1] and

t = Tn(t) :=tn−1 + tn

2+

τn

2t ∈ In ∀ t ∈ I , n = 1, . . . , N. (10)

Let φj ∈ Pk(I), j = 0, . . . , k, denote the basis functions satisfying the conditions

φj(ti) = δi,j, i, j = 0, . . . , k, (11)

where t0 = −1 and ti, i = 1, . . . , k, are the standard Gauß quadrature points for the reference

interval I. Then, we define the basis functions on the original time interval In by

φn,j(t) := φj(t) with t := T−1n (t) =

2

τn

(t− tn − tn−1

2

)∈ I . (12)

Similarly, we define the test basis functions ψn,i by suitable reference basis functions ψi ∈Pk−1(I), i.e.,

ψn,i(t) := ψi(T−1n (t)) ∀ t ∈ In, i = 1, . . . , k. (13)

For practical computations, we have to approximate the right hand side in the exact cGP(k)-

method (5) by some numerical integration. To this end, we replace the function f(t) by the

time-polynomial πkf ∈ Pk(In, L2(Ω)) defined as the Lagrange interpolate

πkf(t) :=k∑

j=0

f(tn,j)φn,j(t) ∀ t ∈ In.




Now, we transform all integrals in (5) to the reference interval I and obtain the following

system of equations for the ”coefficients” U jn ∈ V in the ansatz (6)

k∑

j=0

αi,j

(U jn, v)Ω+

τn

2βi,ja(U

jn, v)

=

τn

2

k∑

j=0

βi,j (f(tn,j), v)Ω ∀ v ∈ V (14)

where i = 1, . . . , k,

αi,j :=

∫

Iφ′j(t)ψi(t) dt, βi,j :=

∫

Iφj(t)ψi(t) dt (15)

and the ”coefficient” U0n ∈ V is known. Due to the polynomial degree of φj and ψi we can

compute the integrals for αi,j and βi,j exactly by means of the k-point Gauß formula with

weights wµ and points tµ, µ = 1, . . . , k. If we choose the test functions ψi ∈ Pk−1(I) in (15)

such that

ψi(tµ) = (wi)−1δi,µ ∀ i, µ ∈ 1, . . . , k,

we get from (15) that

αi,j = φ′j(ti), βi,j = δi,j , 1 ≤ i ≤ k, 0 ≤ j ≤ k.

Then, the system (14) is equivalent to the following coupled system of equations for the

k unknown ”coefficients” U jn ∈ V , j = 1, . . . , k,

k∑

j=0

αi,j

(U jn, v)Ω+

τn

2a(U i

n, v) =τn

2(f(tn,i), v)Ω ∀ v ∈ V, i = 1, . . . , k, (16)

where U0n = uτ(tn−1) for n > 1 and U0

1 = u0. In the following, we specify the cGP(k)-method

for the cases k = 1 and k = 2.

2.1 cGP(1)-method

We use the one-point Gauß quadrature formula with the point t1 = 0 and tn,1 = tn−1 +τn

2 .

Then, we get α1,0 = −1 and α1,1 = 1. Thus, equation (16) leads to the following equation for

the one ”unknown” U1n = uτ(tn−1 +

τn

2 ) ∈ V(U1n, v)Ω+

τn

2a(U1

n, v) =τn

2(f(tn,1), v)Ω +

(U0n, v)Ω

∀ v ∈ V. (17)

Once we have determined the solution U1n, we get the solution at discrete time tn by means

of linear extrapolation

uτ(tn) = 2U1n − U0

n, (18)

where U0n is the initial value at the time interval [tn−1, tn] coming from the previous time

interval or the initial value u0. If we would replace f(tn,1) by the mean value (f(tn−1) +

f(tn))/2, which means that we replace the one-point Gauß quadrature of the right hand side

by the Trapezoidal rule, the resulting cGP(1)-method would be equivalent to the well-known

Crank-Nicolson scheme.




2.2 cGP(2)-method

We use the 2-point Gauß quadrature formula with the points t1 = − 1√3and t2 = 1√

3. Then,

we obtain the coefficients

(αi,j) =

(−√3 3

22√3−32√

3 −2√3−32

32

)i = 1, 2, j = 0, 1, 2.

On the time interval In = [tn−1, tn] we have to solve for the two ”unknowns” U jn = uτ(tn,j)

with tn,j := Tn(tj) for j = 1, 2. The coupled system for U1n, U

2n ∈ V reads

α1,1

(U1n, v)Ω+ τn

2 a(U1n, v)

+ α1,2

(U2n, v)Ω

= τn

2 (f(tn,1), v)Ω − α1,0

(U0n, v)Ω,

α2,1

(U1n, v)Ω+α2,2

(U2n, v)Ω+ τn

2 a(U2n, v)

= τn

2 (f(tn,2), v)Ω − α2,0

(U0n, v)Ω,

(19)

which has to be satisfied for all v ∈ V . Once we have determined the solution (U1n, U

2n), we

get the solution at discrete time tn by means of quadratic extrapolation

uτ(tn) = U0n +

√3(U2

n − U1n), (20)

where U0n is the initial value at the time interval In coming from the previous time interval

or the initial value u0.

3 Space Discretization by FEM

After discretizing equation (1) in time, we now apply the finite element method to discretize

each of the ”In-problem” in space. To this end, let Vh ⊂ V denote a finite element space. In

the numerical experiments, Vh will be defined by biquadratic finite elements on a quadrilateral

mesh Th for the computational domain Ω. Each ”In-problem” for the cGP(k)-method or the

dG(k-1)-method has the structure: For given U0n ∈ V , find U1

n, . . . , Ukn ∈ V such that

k∑

j=1

γi,j(U jn, v)Ω+

τn

2a(U i

n, v) = di(U0n, v)Ω+

τn

2(f(tn,i), v)Ω ∀ v ∈ V, (21)

for all i = 1, . . . , k, where γi,j and di are given constants. In the space discretization, each

U jn ∈ V is approximated by a finite element function U j

n,h ∈ Vh and the fully discrete ”In-

problem” reads: For given U0n,h ∈ Vh, find U

1n,h, . . . , U

kn,h ∈ Vh such that

k∑

j=1

γi,j

(U jn,h, vh

)Ω+

τn

2a(U i

n,h, vh) = di(U0n,h, vh

)Ω+

τn

2(f(tn,i), vh)Ω ∀ vh ∈ Vh, (22)

for all i = 1, . . . , k. Once we have solved this system, we can compute for each time t ∈ In a

finite element approximation uτ,h(t) ∈ Vh of the time discrete solution uτ(t) ∈ V . To this end,




we replace the ”constants” U jn ∈ V in the ansatz of uτ(t) by the space discrete ”constants”

U jn,h ∈ Vh.In the following, we will write the problem (22) as a linear algebraic block system. Let

bµ ∈ Vh, µ = 1, . . . ,mh, denote the finite element basis functions and U jn ∈ R

mh the nodal

vector of U jn,h ∈ Vh such that

U jn,h(x) =

mh∑

µ=1

(U jn)µbµ(x) ∀ x ∈ Ω.

Furthermore, let us introduce the mass matrix M ∈ Rmh×mh , the discrete Laplacian matrix

L ∈ Rmh×mh and the vector F i

n ∈ Rmh as

Mν,µ := (bµ, bν)Ω , Lν,µ := a(bµ, bν), (F in)ν := (f(tn,i), bν)Ω . (23)

Then the fully discrete ”In-problem” is equivalent to the following linear k× k block system:

For given U0n ∈ R

mh, find U1n, . . . , U

kn ∈ R

mh such that

k∑

j=1

γi,jMU jn +

τn

2LU i

n = diMU0n +

τn

2F in, ∀ i = 1, . . . , k. (24)

The vector U0n is defined as the finite element nodal vector of the fully discrete solution

uτ,h(tn−1) computed from the previous time interval [tn−2, tn−1] if n ≥ 2 or from a finite

element interpolation of the initial data u0 if n = 1.

In the following, we will present the resulting block systems for the cGP(1), cGP(2) and

dG(1) method which are used in our numerical experiments.

3.1 cGP(1)-method

The problem on time interval In reads: For given U0n ∈ R

mh , find U1n ∈ R

mh such that(M +

τn

2A)U1

n =τn

2F 1n +MU 0

n, (25)

where A = L+K +M denotes the sum of Laplacian, convection and mass matrices, respec-

tively. Once we have determined the solution U1n, we compute the nodal vector U0

n+1 of the

fully discrete solution uτ,h at the time tn by using the following linear extrapolation

uτ,h(tn) ∼ U0n+1 = 2U1

n − U0n.

3.2 cGP(2)-method

The 2× 2 block system on time interval In reads: For given U0n ∈ R

mh, find U1n, U

2n ∈ R

mh

such that (3M + τnA

(2√3− 3

)M

(−2

√3− 3

)M 3M + τnA

)(U1

n

U2n

)=

(R1

n

R2n

)(26)




where A = L+K +M denotes the sum of Laplacian, convection and mass matrices, respec-

tively and

R1n = τnF

1n + 2

√3MU0

n

R2n = τnF

2n − 2

√3MU0

n.

Once we have determined the solution (U1n, U

2n), we compute the nodal vector U0

n+1 of the

fully discrete solution uτ,h at the time tn by using the following quadratic extrapolation

uτ,h(tn) ∼ U0n+1 = U0

n +√3(U2

n − U1n).

4 The nonconforming Qn3-element

A family of higher order nonconforming quadrilateral finite elements (Qr-elements) based

on a new compatiblity condition was presented in [8]. It was shown in [6] that on general

unstructured grids, which are not multilevel grids, the order of approximation can be non-

optimal. To overcome this drawback, it has been proposed in [6] to work either with a new

Qbr-element, which includes additional bubble function, or with the Qn

r -element based on

the so-called nonparametric approach of [9] which does not use the usual bilinear reference

mapping for the definition of the basis functions.

In this paper, we choose the latter approach and work with the Qn3 -element which is of

fourth order accurate in the L2-norm. We will shortly describe this element in the rest of this

section. For a given mesh cell K ∈ Th, let TK : K → K denote the usual bilinear mapping

on the reference element K := (−1, 1)2. We linearize the mapping TK in the barycenter of K

and obtain an affine mapping TK defined by

TK(x, y) := DTK |(0,0) · (x, y)⊤ + TK(0, 0). (27)

Furthermore, let K := T−1K (K) denote the pre-image of the element K under TK , see Figure 1,

such that the original mesh cell K is the affine image of the ”modified reference element” K.

K K KTK T−1

K

Figure 1: Correlation between the reference element K, the real element K and K.

If the shape of K is close to that of a parallelogram then K is close to K. The basic idea

of the nonparametric approach is to construct the generating local basis functions ϕk as




polynomials on K (and not on K). Thus, the ”real” local basis functions ϕi : K → R defined

as ϕi(x, y) := ϕi

(T−1K (x, y)

)are polynomials in the real world coordinates (x, y). In the

following, let Lj and Lj,k denote the 1D and 2D Legendre polynomials

L0(s) := 1, L1(s) := s, L2(s) :=1

2(3s2 − 1), Lj,k(x, y) := Lj(x) · Lk(y). (28)

Then, the following fifteen nodal functionals are associated with the Qn3 -element:

• For j, k ∈ 0, 1, j + k ≤ 1, the three cell moments

N Kj,k(ϕ) := |K|−1

∫

Kϕ(x, y) · Lj,k(x, y) dK. (29)

• For i ∈ 1, . . . , 4 and j ∈ 0, 1, 2 the twelve edge moments

N Ei

j (ϕ) := |Ei|−1

∫

Ei

ϕ(x, y) ·(Lj T−1

Ei

)(x, y) dEi (30)

of the four associated edges Ei of K, with TEi: (−1, 1) → Ei denoting the affine

parameterization of Ei.

For the case K = K, it was shown in [4] that the space

V3 := span1, x, y, x2, xy, y2, x3, x2y, xy2, y3, x2y2,

x3y2, x2y3, x3y − xy3, x4y2 − x2y4

(31)

is unisolvent with respect to the above set of nodal functionals. Let p1, . . . , p15 denote the

monomial basis functions of V3 given in (31), N1, . . . , N15 the nodal functionals defined in (29)

and (30) and C ∈ R15×15 the matrix defined by

Cij := Ni(pj). (32)

Assuming that V3 is unisolvent with respect to the Ni also for the actual modified reference

element K := T−1K (K), we get that C is regular, that means that its inverse C−1 exsits. Then,

the local basis functions ϕi : K → R, i = 1, . . . , 15, that satisfy the usual duality property

Nj(ϕi) = δi,j ∀ i, j ∈ 1, . . . , 15, (33)

can be computed by

ϕi(x, y) :=15∑

j=1

(C−1)ij · pj(x, y). (34)

The unisolvence for the general case K 6= K can be guaranteed if the unstructured mesh Thdoes not have quadrilaterals that are too much distorted. The local basis functions ϕi : K → R




on the original mesh cell K ∈ Th are defined by means of the affine mapping TK : K → K as

ϕi := ϕi T−1K . Due to the choice of V3, we have that the polynomial space P3(K) is contained

in V3. Since the mapping TK is affine we can show that

P3(K) ⊂ V3 := spanϕ1, . . . , ϕ15

, (35)

which ensures the optimal approximation order of the Qn3 -element, see [1].

5 Solution of the linear systems

The resulting discretized linear systems in each time interval [tn−1, tn], which are, for the

cGP(1)-method, 1x1 block systems of the form

(M +

τn

2A)U1

n =τn

2F 1n +MU 0

n, (36)

and, for the cGP(2)-method, 2x2 block systems of the form

(3M + τnA

(2√3− 3

)M

(−2

√3− 3

)M 3M + τnA

)(U1

n

U2n

)=

(R1

n

R2n

)(37)

are treated by using a geometrical multigrid solver with corresponding (block) smoothers

and grid transfer operators. In this paper, we use the standard refinement scheme (see [11])

for the grid hierarchies, and for the smoothing operator, we choose the standard (pointwise)

Gauß-Seidel method (with four pre- and post-smoothing steps); however, also corresponding

standard (block) variants of Jacobi, SOR and ILU methods can be easily applied, too. More-

over, we use for the canonical grid transfer routines the standard FEM projection operator

defined for the Qn3 -element (see [11] and [7] for a corresponding approach for first order non-

conforming and biquadratic conforming finite elements). Finally, the coarse grid problem is

solved by a direct solver.

6 Numerical results

In this section, we perform several numerical tests to analyze the presented spatial and tem-

poral discretizations. For all examples, we use the domain Ω = (0, 1)2 and the time interval

I = [0, T ] with T = 1. In order to be able to measure the exact error of the numerical solution

we prescribe an anlytical exact solution u(x, y, t) and compute the associated data f and u0

from the equation (1).




6.1 Error in space

For the study of the approximation properties of the Qn3 -element, we consider sequences of

structured and semi-structured meshes which are generated by uniform refinement from a

coarsest mesh with one and three quadrilateral mesh cells, respectively. Starting from the

coarsest grid defined as mesh level ℓ = 1, we generate the grid of mesh level ℓ+1 by dividing

each quadrilateral cell of grid level ℓ into four new quadrilaterals connecting the midpoints

of opposite edges. Figure 2 shows the grids on level ℓ = 1, 2, 3 for the structured and semi-

structured meshes. In case of the structured mesh, the mesh size on grid level ℓ is h = 2−ℓ+1.

Figure 2: Structured (above) and semi-structured (below) grids on space mesh

level=1,2,3 (from left to right).

Example 1 We consider the stationary Convection-Diffusion-Reaction equation with homo-

geneous Dirichlet boundary conditions. We prescribe the smooth exact solution in our model

problem as

u(x, y, t) := sin(πx) sin(πy),

and the associated data f . The parameters are set to α = 1, β = (1, 1) and γ = 1.

To analyze the quality of the spatial discretization, we demonstrate in Table 1 and 2 the

behavior of the standard L2/H1-error norms for different refinement levels. By ”EOC” we

denote the experimental order of convergence and in the column ”#DOFs” we show the total




L2-error H1-error DOFs CPU-time

Lev ‖u− uh‖2 EOC ‖∇u−∇uh‖2 EOC #dofs Factor CPU Factor

3 3.60E-04 1.22E-02 168 0.01

4 2.30E-05 3.97 1.56E-03 2.97 624 3.71 0.03 3.76

5 1.45E-06 3.99 1.96E-04 2.99 2400 3.85 0.11 3.99

6 9.05E-08 4.00 2.45E-05 3.00 9408 3.92 0.44 4.05

7 5.66E-09 4.00 3.06E-06 3.00 37248 3.96 1.79 4.05

8 3.54E-10 4.00 3.82E-07 3.00 148224 3.98 7.52 4.19

9 2.23E-11 3.99 4.78E-08 3.00 591360 3.99 31.37 4.17

Table 1: Discretization error of the Q3-element in the L2-norm and theH1-norm on structured

meshes.

L2-error H1-error DOFs CPU-time

Lev ‖u− uh‖2 EOC ‖∇u−∇uh‖2 EOC #dofs Factor CPU Factor

3 9.96E-05 4.84E-03 468 0.02

4 6.50E-06 3.94 6.26E-04 2.95 1800 3.85 0.07 3.79

5 4.09E-07 3.99 7.87E-05 2.99 7056 3.92 0.30 4.49

6 2.56E-08 4.00 9.84E-06 3.00 27936 3.96 1.42 4.65

7 1.60E-09 4.00 1.23E-06 3.00 111168 3.98 6.67 4.71

8 1.00E-10 4.00 1.53E-07 3.00 443520 3.99 31.97 4.79

9 8.24E-12 3.60 1.92E-08 3.00 1771776 3.99 140.54 4.40

Table 2: Discretization error of the Q3-element in the L2-norm and the H1-norm on semi-

structured meshes.

number of all unknowns of the 2x2 block-systems on each time interval which is 2mh where

mh denotes the dimension of the space Vh. Furthermore, we present in column ”CPU” the

CPU-times in seconds for the linear solver implemented within the solver package FEAT2

(www.featflow.de) and performed on a Dual-Core AMD Opteron 8220 with eight CPUs at

2.8GHz.

Table 1 and 2 show that the L2-norm of the error behaves like order O(h4) and the H1-

error by order O(h3) for mesh size h as expected. Moreover, it can be seen that this element

works well on the structured and the semi-structured meshes. Comparing the computational

cost for the linear solver, we observe that the CPU-time on the next higher grid level ℓ + 1

increases by a factor of appr. 4 whereas the corresponding number of degrees of freedom grows

by factor of 4. This shows that our coupled multigrid solver has nearly optimal computational

complexity.




6.2 Error in time

In this section, we perform numerical tests in order to analyze the accuracy of the proposed

time discretization schemes. For all tests, we use an equidistant time step size τ = T/N where

T = 1. The discretization error u − uτ,h over the time interval I = [0, 1] is measured in the

standard L2-norm ‖ · ‖2,L := ‖ · ‖L2(I,L2(Ω)) and the discrete L∞-seminorm defined as

|v|∞,L := max1≤n≤N

‖v(tn)‖L2(Ω), tn := nτ. (38)

Example 2 As a test example we consider problem (1) with the right hand side derived from

the prescribed exact solution

u(x, y, t) = x(1− x)y(1 − y)et.

The initial data is u0(x, y) = u(x, y, 0).

This example has the property that there is no spatial error which can be seen as follows.

For each fixed time t, the exact solution u(t) is an element of the finite element space Vh.

Therefore, the standard semi-discrete solution with respect to space uh(t) ∈ Vh is equal to u(t).

If we now apply the time discretization to uh(t) = u(t) we get uh,τ(t) = uτ(t). Since space

and time discretizations are of Galerkin type we obtain uτ,h(t) = uh,τ(t) = uτ(t). Therefore,

the full discretization error u(t)−uτ,h(t) is equal to the time discretization error u(t)−uτ(t),

i.e. we will concentrate only on the time discretization error and exclude interactions with

the spatial error.

We apply the time higher order time discretization scheme cGP(2) with an equidistant

time step size τ = T/N . To measure the error, the following discrete time L∞-norm of a

function v : I → L2(Ω) is used

‖v‖∞ := max1≤n≤N

‖v−(tn)‖L2(Ω), v−(tn) := limt→tn−0

v(t), tn := nτ.

The behavior of the standard L2-norm ‖·‖2 := ‖·‖L2(I,L2(Ω)) and the discrete L∞-norm of the

time discretization error u(t)− uτ(t) over the time interval I = [0, 1] can be seen in Table 3.

The estimated value of the experimental order of convergence (EOC) is also calculated and

compared with the theoretical order of convergence. Here, we set the diffusion, convection

and reaction coefficients is equal to 1.

We see that the cGP(2)-method is of order 3 in the L2-norm and superconvergent of

order 4 at the discrete points tn, as expected from the theory. Next, we want to show the

influence of edge oriented jump (EOJ) stabilization on the accuracy of these time discretization

schemes. To this end, we perform some numerical tests for the high order cGP(2)-method




together with Q3-element being applied with edge oriented jump (EOJ) stabilization. The

stabilization parameter is set to 0.01 in our numerical results. From Table 4, it can be seen

that both the temporal and spatial discretizations works well with the edge oriented jump

stabilization and confirm their orders of convergence.

cGP(2)

1/τ ‖u− uτ‖2 EOC ‖u− uτ‖∞ EOC

10 4.20E-07 4.41E-07

20 5.12E-08 3.04 2.83E-08 3.96

40 6.35E-09 3.01 1.78E-09 3.99

80 7.92E-10 3.00 1.12E-10 3.99

160 9.90E-11 3.00 6.95E-12 4.01

320 1.24E-11 3.00 4.22E-13 4.04

640 1.55E-12 3.00 2.85E-14 3.89

1280 1.97E-13 2.97

Table 3: Error norms for the Example 2, for cGP(2)-method without EOJ stabilization.

cGP(2)

1/τ ‖u− uτ‖2 EOC ‖u− uτ‖∞ EOC

10 4.21E-07 4.41E-07

20 5.14E-08 3.03 2.84E-08 3.96

40 6.37E-09 3.01 1.78E-09 3.99

80 7.92E-10 3.01 1.13E-10 3.98

160 1.00E-10 2.99 6.96E-12 4.02

320 1.25E-11 3.00 4.23E-13 4.04

640 1.55E-12 3.01 2.85E-14 3.89

1280 1.97E-13 2.97

Table 4: Error norms for the Example 2, for cGP(2)-method with EOJ stabilization.

6.3 Error in space and time

Now, we analyze the behavior of the full discretization error u(t) − uh,τ(t) in an example

where an approximation error in space as well as in time occurs.

Example 3 We consider problem (1) with the prescribed exact solution

u(x, y, t) := sin(πx) sin(πy) sin(10πt),




and the associated data f and u0.

In Table 5, we present in each of the three column blocks for grid level ℓ = 6, 7, 8, the error

norms and the experimental orders of convergence (EOC) for decreasing τ. For a fixed mesh

size hℓ = 2−(ℓ−1) and τ → 0, the space error becomes dominant for sufficiently small time

step sizes τ < τ0(hℓ). We indicate by means of an underline that row in the column block

of grid level ℓ which corresponds to the last suitable time step size τ0(hℓ). Whereas Table 5

shows the results for the structured meshes we show in Table 6 the analogous results for the

semi-structured meshes.

ℓ=6 ℓ=7 ℓ=8

1/τ ‖u− uh,τ‖2,L EOC ‖u− uh,τ‖2,L EOC ‖u− uh,τ‖2,L EOC

10 3.51E-02 3.51E-02 3.51E-02

20 9.37E-03 1.90 9.37E-03 1.90 9.37E-03 1.90

40 1.17E-03 3.00 1.17E-03 3.00 1.17E-03 3.00

80 1.46E-04 3.00 1.46E-04 3.00 1.46E-04 3.00

160 1.82E-05 3.00 1.82E-05 3.00 1.82E-05 3.00

320 2.28E-06 3.00 2.28E-06 3.00 2.28E-06 3.00

640 2.92E-07 2.96 2.85E-07 3.00 2.85E-07 3.00

1280 7.32E-08 1.99 3.58E-08 2.99 3.56E-08 3.00

2560 6.42E-08 0.19 5.98E-09 2.58 4.45E-09 3.00

1/τ |u− uh,τ|∞,L EOC |u− uh,τ|∞,L EOC |u− uh,τ|∞,L EOC

10 4.27E-02 4.27E-02 4.27E-02

20 3.18E-03 3.75 3.18E-03 3.75 3.18E-03 3.75

40 1.99E-04 4.00 1.99E-04 4.00 1.99E-04 4.00

80 1.24E-05 4.00 1.24E-05 4.00 1.24E-05 4.00

160 7.84E-07 3.98 7.83E-07 3.99 7.83E-07 3.99

320 9.15E-08 3.10 4.90E-08 4.00 4.90E-08 4.00

640 9.05E-08 0.01 5.72E-09 3.10 3.07E-09 4.00

1280 9.05E-08 0.00 5.66E-09 0.01 3.57E-10 3.10

2560 9.05E-08 0.00 5.66E-09 0.00 3.54E-10 0.01

Table 5: Full discretization error u − uh,τ measured in ‖ · ‖2,L and | · |∞,L for Example 3 at

different levels of structured space meshes.

We see that in the standard L2-norm ‖ · ‖2,L the error of the cGP(2)-method behaves, for

fixed mesh size hℓ, like O(τ3) as long as τ ≥ τ0(hℓ) whereas it starts to stagnate for τ < τ0(hℓ).

If we look at the error norms for the sequence of space-time meshes with (τ, h) = (τ0(hℓ), hℓ),

ℓ = 6, 7, 8, we observe that the error decreases by a factor of about 16 if we increase the level




ℓ=6 ℓ=7 ℓ=8

1/τ ‖u− uh,τ‖2,L EOC ‖u− uh,τ‖2,L EOC ‖u− uh,τ‖2,L EOC

10 3.51E-02 3.51E-02 3.51E-02

20 9.37E-03 1.90 9.37E-03 1.90 9.37E-03 1.90

40 1.17E-03 3.00 1.17E-03 3.00 1.17E-03 3.00

80 1.46E-04 3.00 1.46E-04 3.00 1.46E-04 3.00

160 1.82E-05 3.00 1.82E-05 3.00 1.82E-05 3.00

320 2.28E-06 3.00 2.28E-06 3.00 2.28E-06 3.00

640 2.85E-07 3.00 2.85E-07 3.00 2.85E-07 3.00

1280 3.99E-08 2.84 3.56E-08 3.00 3.56E-08 3.00

2560 1.86E-08 1.10 4.59E-09 2.96 4.45E-09 3.00

1/τ |u− uh,τ|∞,L EOC |u− uh,τ|∞,L EOC |u− uh,τ|∞,L EOC

10 4.27E-02 4.27E-02 4.27E-02

20 3.18E-03 3.75 3.18E-03 3.75 3.18E-03 3.75

40 1.99E-04 4.00 1.99E-04 4.00 1.99E-04 4.00

80 1.24E-05 4.00 1.24E-05 4.00 1.24E-05 4.00

160 7.83E-07 3.99 7.83E-07 3.99 7.83E-07 3.99

320 4.92E-08 3.99 4.90E-08 4.00 4.90E-08 4.00

640 2.56E-08 0.94 3.08E-09 3.99 3.06E-09 4.00

1280 2.56E-08 0.00 1.60E-09 0.94 1.94E-10 3.98

2560 2.56E-08 0.00 1.60E-09 0.00 1.03E-10 0.92

Table 6: Full discretization error u − uh,τ measured in ‖ · ‖2,L and | · |∞,L for Example 3 at

different levels of semi-structured space meshes.

ℓ by one. This indicates an asymptotic behaviour of the form

‖u− uτ,h‖2, L ≤ C(τ3 + h4),

while in case of the | · |∞,L, the asymptotic behaviour is

|u− uτ,h|∞,L ≤ C(τ4 + h4).

This asymptotic behaviour is optimal with respect to the quadratic polynomial ansatz of the

cGP(2)-method in time and the cubic ansatz of the Qn3 -element in space. Table 6 shows

analogous results for the case of semi-structured meshes. Moreover, it demonstrates that the

temporal accuracy is not disturbed due to the semi-structured meshes.

6.4 Solver Analysis

Next, we perform numerical tests to analyze the behavior of the corresponding multigrid

solver in the cGP(2)-method for Example 3. In order to analyze the multigrid behavior, we




present in Table 7 the averaged number of multigrid iterations per time step for solving the

corresponding systems. The solver stops if the L2-norm of the relative residual is smaller than

10−10 or the absolute residual drops down by 10−15. From Table 7, we see that, for fixed time

ℓ τ = 1/10 τ = 1/20 τ = 1/40 τ = 1/80 τ = 1/160 τ = 1/320 τ = 1/640

3 11.0 10.0 9.0 8.9 8.0 7.0 7.0

4 12.0 11.5 11.5 11.0 9.1 9.0 8.0

5 12.0 12.0 12.0 11.9 11.9 11.0 9.0

6 12.0 12.0 12.0 12.0 12.0 12.0 11.8

7 12.0 12.0 12.0 12.0 12.0 11.8 11.5

8 12.0 12.0 12.0 12.0 11.8 11.4 10.9

3 13.0 12.5 12.0 11.0 10.0 9.9 9.0

4 14.0 14.0 14.0 13.9 13.0 12.0 10.0

5 16.0 16.0 15.8 15.3 15.1 14.9 13.9

6 17.0 17.0 17.0 17.0 16.0 15.8 15.5

7 17.0 17.0 17.0 16.8 16.5 15.6 14.6

8 17.0 17.0 17.0 16.0 15.4 14.4 14.0

Table 7: Averaged multigrid iterations per time step for the cGP(2)-method on structured

(above) and semi-structured (below) meshes.

step τ, the multigrid solver requires almost the same number of iterations for increasing grid

level ℓ. Moreover, if we decrease τ for fixed grid level ℓ, the number of multigrid iterations

does not increase. This means that the behavior of the multigrid solver is almost independent

of the space mesh size and the time step. Comparing the convergence behavior between

structured and semi-structured meshes, we see that the averaged number of iterations is only

slightly higher on semi-structured meshes but it remains also almost independent of the space

mesh size and the time step.

6.5 Comparison with low order scheme

Next, for Example 3 and on structured meshes, we compare the time discretization schemes

cGP(2) and the cGP(1)- (or Crank-Nicolson) method concerning the achieved accuracy and

the required numerical cost. Table 8 shows, for different τ and grid level ℓ = 6, the global

L∞-norm error and the total CPU-time (in seconds) required for the computations on all time

intervals. One can see that, in order to achieve the accuracy of 10−6, we need the very small

time step τ = 1/10240 for the cGP(1)(or Crank-Nicolson) scheme while the same accuracy has

be already achieved with τ = 1/160 in the case of the cGP(2)-scheme, due to superconvergent

results of 4th order accuracy in the discrete time points. Thus, for a desired accuracy of 10−6,




cGP(1) cGP(2)

1/τ |u− uh,τ|∞,L CPU |u− uh,τ|∞,L CPU

10 2.35E-15 0.47 4.27E-02 17.02

20 8.79E-02 12.76 3.18E-03 38.33

40 2.25E-02 21.66 1.99E-04 75.45

80 5.59E-03 47.47 1.24E-05 151.47

160 1.42E-03 83.91 7.84E-07 307.25

320 3.54E-04 194.63

640 8.84E-05 332.36

1280 2.21E-05 718.33

2560 5.52E-06 1213.70

5120 1.38E-06 2468.65

10240 3.54E-07 3859.79

Table 8: Error norms |u− uh,τ|∞,L and total CPU-times to achieve the accuracy of 10−6 on

grid level ℓ = 6.

the scheme cGP(2) is about 12 times faster than cGP(1)(or Crank-Nicolson)-scheme.

Finally, we compare the accuracy and the numerical cost of our proposed higher order

space-time discretization scheme cGP(2)-Qn3 with the lower order schemes and cGP(1)- (or

Crank-Nicolson)-Q1, respectively, where Qn3 , and Q1 indicate the used finite element spaces

Vh which are the nonparametric nonconforming space of order 4 and the usual conforming

space of bilinear elements. To this end, we present in each of the Tables 9–10 the full dis-

cretization error u − uh,τ of the corresponding scheme in the discrete L∞-norm, the total

number ”#DOFs” of all unknowns occurring on the space mesh (i.e. on each time interval)

and the required CPU-time in seconds where the mesh and time step size have been chosen

as

h = 2−(ℓ−1), τ =1

52−ℓ, ℓ = 2, 3, . . . .

For the scheme cGP(2)-Qn3 , it can be seen from Table 9 that the full discretization error is

reduced by a factor of 16 if we increase the level ℓ by one.

That means we gain a factor of 16 in accuracy whereas the numerical cost in terms of

the CPU-time increase only by a factor of 8 which is a big advantage for the higher order

method. For the scheme cGP(1)-Q1 we see in 10, that we gain only a factor of 4 whereas the

CPU-time increases by a factor of 8.

If we ask, for instance, for a discrete solution with an accuracy of 1.46 10−6 then we would

need 41.20 s with the scheme cGP(2)-Qn3 , and 19692.14 s with the cGP(1)-Q1-method (see

the underlined values in Table 10–12), i.e., the cGP(2)-Qn3 -scheme is 480 times faster than the

Crank-Nicolson scheme with bilinear elements. The acceleration factors in the CPU-time of




ℓ 1/τ |u− uh,τ|∞,L Factor #DOFs Factor CPU Factor

2 20 5.37E-03 96 0.08

3 40 3.64E-04 14.77 336 3.50 0.58 7.66

4 80 2.32E-05 15.67 1248 3.71 5.03 8.64

5 160 1.46E-06 15.92 4800 3.85 41.20 8.19

6 320 9.15E-08 15.94 18816 3.92 335.65 8.15

7 640 5.72E-09 16.00 74496 3.96 2735.65 8.15

8 1280 3.59E-10 15.95 296448 3.98 22335.65 8.16

Table 9: Full discretization error, total number of unknowns and CPU-time for the cGP(2)-Qn3

scheme.

ℓ 1/τ |u− uh,τ|∞,L Factor #DOFs Factor CPU Factor

2 20 1.32E-01 9 0.01

3 40 3.11E-02 4.25 25 0.03 4.00

4 80 7.64E-03 4.06 81 3.24 0.19 5.94

5 160 1.90E-03 4.02 289 3.57 0.99 5.19

6 320 4.75E-04 4.00 1089 3.77 5.76 5.84

7 640 1.19E-04 4.00 4225 3.88 39.51 6.86

8 1280 2.97E-05 4.00 16641 3.94 305.11 7.72

9 2560 7.43E-06 4.00 66049 3.97 2450.89 8.03

10 5120 1.86E-06 4.00 263169 3.98 19692.14 8.03

11 10240 4.64E-07 4.00 1050625 3.99 160964.07 8.17

Table 10: Full discretization error, total number of unknowns and CPU-time for the cGP(1)-

Q1 scheme.

the higher order method compared to the lower order ones are growing if we ask for a higher

accuracy of the discrete solution. This can be seen in Figure 3 where we have plotted for all

three methods the CPU-time against the accuracy.

That means we gain a factor of 16 in accuracy whereas the numerical cost in terms of

the CPU-time increase only by a factor of 8 which is a big advantage for the higher order

method. For the scheme cGP(1)-Q1 we see in 10, that we gain only a factor of 4 whereas the

CPU-time increases by a factor of 8.

7 Conclusion

We have presented continuous Galerkin-Petrov time discretization schemes for the two di-

mensional Convection-Diffusion-Reaction equation where the spatial discretization has been

performed by means of cubic finite elements. In particular, we have analyzed the 4th order




10−10

10−8

10−6

10−4

10−2

100

10−2

10−1

100

101

102

103

104

105

106

||u−uh,τ||∞

CP

U−t

ime

cGP(1)-Q1

cGP(2)-Qn

3

Figure 3: Full discretization error |u− uh,τ|∞,L vs. required CPU-time.

cGP(2)-method, in combination with 4th order nonconforming Qn3 -finite element approxima-

tion in space for the numerical solution of the two dimensional Convection-Diffusion-Reaction

equation. Moreover, the space discretization is also combined with edge oriented jump sta-

bilization (EOJ) which is required for the increasing convection parameter. The associated

linear systems have been solved using a geometrical multigrid method. Firstly, from the nu-

merical studies, we observe that the full discretization error decreases by a factor of 16 with

respect to the space mesh size and time step h and τ, respectively. Moreover, the estimated

experimental orders of convergence confirm the theoretical orders with and without edge

oriented jump stabilization. Furthermore, the tests show that the cGP(2)-scheme provides

significantly more accurate numerical solutions than the other presented schemes cGP(1) (or

Crank-Nicolson) which means that quite large time step sizes are allowed to gain highly ac-

curate results. Secondly, we can see that the used multigrid solver is much more efficient

since it shows a robust convergence behavior which is nearly independent of the space mesh

size and the time step such that large time steps get feasible also with respect to efficient

solution methods. In our future work, we plan to extend these time discretization schemes

to the Navier-Stokes equations to simulate complex time dependent flow problems in a very




efficient way together with highly sophisticated multigrid-FEM techniques for incompressible

saddle point problems

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Galerkin discretizations of 4th order in space and time ... · obtain such superconvergence results...

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