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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
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:04 2
131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
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
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:04 3
131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
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)
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131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
4 R. Mahmood, S. Hussain
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)
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131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
Galerkin time discretizations of 4th order in space and time 5
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.
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6 R. Mahmood, S. Hussain
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.
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131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
Galerkin time discretizations of 4th order in space and time 7
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,
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131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
8 R. Mahmood, S. Hussain
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)
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Galerkin time discretizations of 4th order in space and time 9
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
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10 R. Mahmood, S. Hussain
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
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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).
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12 R. Mahmood, S. Hussain
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
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Galerkin time discretizations of 4th order in space and time 13
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.
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14 R. Mahmood, S. Hussain
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
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Galerkin time discretizations of 4th order in space and time 15
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),
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16 R. Mahmood, S. Hussain
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
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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
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18 R. Mahmood, S. Hussain
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,
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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
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20 R. Mahmood, S. Hussain
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
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:04 21
131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
Galerkin time discretizations of 4th order in space and time 21
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
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:04 22
131104-6767- IJBAS-IJENS @ August 2013 IJENS I J E N S
22 R. Mahmood, S. Hussain
efficient way together with highly sophisticated multigrid-FEM techniques for incompressible
saddle point problems
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