Estimating the error of a H 1 -mixed finite element solution for the Burgers equation

ANZIAM J. 56 (CTAC2014) pp.C383–C398, 2016 C383

Estimating the error of a H ¹ -mixed finite element solution for the Burgers equation

Sabarina Shafie ¹ Thanh Tran ²

(Received 27 February 2015; revised 12 January 2016)

Abstract

We compute error estimations for a H

¹

-mixed finite element method for Burgers equation. By using a H

¹

-mixed finite element method, the problem is reformulated as a system of first order partial differential equations, which allows an approximation of the unknown function and its derivative. Local parabolic and elliptic methods approximate the true errors from the computed solutions; the so-called a posteriori error estimates. Numerical experiments show that the error estimations converge to the true errors.

http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/9356

gives this article, c Austral. Mathematical Soc. 2016. Published February 17, 2016, as

part of the Proceedings of the 17th Biennial Computational Techniques and Applications

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this article.

Contents C384

Contents

1 Introduction C384

2 The H

¹

-mixed finite element method C385 3 A posteriori error estimates and implementation issues C390

4 Numerical results C394

5 Conclusion C396

References C396

1 Introduction

We consider the Burgers equation

∂

_t

u(x, t) − ν∂

_xx

u(x, t) + u(x, t)∂

_x

u(x, t) = 0 , x ∈ Ω , t ∈ (0, T ] , (1) with boundary and initial conditions

u(0, t) = u(1, t) = 0 , t ∈ [0, T ] , (2)

u(x, 0) = u

0

(x) , x ∈ Ω , (3)

where ∂

_t

:= ∂/∂t , ∂

_x

:= ∂/∂x , ∂

_xx

:= ∂

²

/∂x

²

, T and ν (viscosity coefficient) are positive constants and Ω := (0, 1) [2].

The aim is to design methods to compute a posteriori error estimations when the solution of (1)–(3) is approximated by a H

¹

-mixed finite element method (H

¹

-mfem).

Using the H

¹

-mfem, the problem is reformulated as a system of first order

partial differential equations, which allows an approximation for u and its

2 The H

¹

-mixed finite element method C385

derivative ∂

_x

u. The H

¹

-mfem considered in this article is based on an approach suggested by Pani for nonlinear parabolic equations [3]. Pany et al. [4] adapted the method to Burgers equation. Section 2 gives details of this H

¹

-mfem.

The method considered in this article is closely related to least squares mixed finite element methods in that the second order partial differential equation is reformulated as a system of first order partial differential equations with a new unknown defined as the flux [7, 8, 9, 10, and references therein].

A posteriori error estimates are a fundamental component in the design of efficient adaptive algorithms for solving partial differential equations. In this study we consider an implicit type of a posteriori error estimation which is based on the procedure developed by Adjerid et al. [1] for one dimensional parabolic systems. This a posteriori error estimation with finite element methods of lines was studied for one dimensional nonlinear parabolic system and the Sobolev equation [5, 6]. For the approximation of the solution we use a mixed formulation of finite element methods of lines with an a posteriori error estimation computed using the procedure developed by Adjerid et al. [1]. To the best of our knowledge, this is the first time this a posteriori error estimation method is considered for the Burgers equation, where the approximate solution is computed using H

¹

-mfem.

2 The H ¹ -mixed finite element method

Throughout this article, h·, ·i and k·k

_H0(Ω)

denote the inner product and norm in H

⁰

(Ω) = L

²

(Ω) , respectively. As usual, the Sobolev space H

¹

(Ω) consists of functions u for which

kuk

_H1(Ω)

= q

kuk

²_H0(Ω)

+ k∂

_x

uk

²_H0(Ω)

exists. The space H

¹₀

(Ω) contains all functions in H

¹

(Ω) with zero trace at

the endpoints of the domain Ω, namely at x = 0, 1 . For any p ∈ [0, ∞] and

2 The H

¹

-mixed finite element method C386

any normed vector space X, L

^p

(X) is the space L

^p

(0, T ; X) of all functions defined in [0, T ] with values in X. The norm in this space k·k

_Lp(X)

is defined as usual. We write L

^p

(L

^∞

) and L

^p

(H

¹

) instead of L

^p

(L

^∞

(Ω)) and L

^p

(H

¹

(Ω)), respectively.

By H

¹

-mfem, ( 1) is reduced to a system of first order equations using a new variable defined as v = u

_x

. As a consequence, (1) is reformulated as

∂

_x

u = v , (4)

∂

_t

u − ν∂

_x

v + uv = 0 . (5)

We multiply (4) by ∂

_x

χ and (5) by −∂

_x

w, where χ ∈ H

¹₀

(Ω) and w ∈ H

¹

(Ω) are arbitrary test functions. Then, using integration by parts and applying the boundary conditions (2), we obtain a weak formulation of (1)–(3): given u

₀

∈ H

¹₀

(Ω) , find (u, v) : [0, T ] → H

¹0

(Ω) × H

¹

(Ω) satisfying, for t > 0 ,

h∂

_x

u(t), ∂

_x

χi = hv(t), ∂

_x

χi for all χ ∈ H

¹₀

(Ω) , (6) h∂

_t

v(t), wi + ν h∂

_x

v(t), ∂

_x

wi = hu(t)v(t), ∂

_x

wi for all w ∈ H

¹

(Ω) , (7) and, for t = 0 ,

hv(0), wi = h∂

_x

u

₀

, wi for all w ∈ H

¹

(Ω) . (8) Remark 1. Existence and uniqueness of the solution of (6)–(8) can be shown using the method of compactness [12, 11]. We will present this result in a future paper.

Remark 2. If u ∈ W

_∞¹

(0, T ; H

¹₀

(Ω)) ∩ L

^∞

(0, T ; H

²

(Ω)) and (u, v) satisfies (6)–

(7) then (u, v) satisfies (4)–(5). Indeed, by using integration by parts we deduce from (6) that ∂

x

(v − ∂

x

u) = 0 which implies

v(x, t) = ∂

_x

u(x, t) + g(t) , (9) for some function g depending on t. Integrating (9) over Ω, noting (8), we infer g(0) = 0 . Also, it follows from (9) and (7) (with w = 1) that

Z

Ω

[∂

_tx

u + g

⁰

(t)] dx = 0 , (10)

2 The H

¹

-mixed finite element method C387

implying g

⁰

(t) = 0 . Hence g ≡ 0 , that is (u, v) satisfies (4). This immediately gives (5).

Solutions to (6)–(8) are approximated using a high order finite element method defined as follows. We first partition the interval Ω into 0 = x

₁

< x

₂

< · · · <

x

_N+1

= 1 , and define h

_l

:= x

_l+1

− x

_l

for l = 1, . . . , N and h := max

_l

h

_l

. The hat function on (x

l−1

, x

l+1

) for l = 2, . . . , N is defined as

φ

_l,1

(x) =

 



 

(x − x

_l−1

)/h

_l−1

, x ∈ [x

_l−1

, x

_l

) , (x

_l+1

− x)/h

_l

, x ∈ [x

_l

, x

_l+1

) ,

0, otherwise.

At the endpoints of Ω (namely, at x = 0, 1) we define

φ

_1,1

(x) =



(x

₂

− x)/h

₁

, x ∈ [x

₁

, x

₂

) ,

0, otherwise,

φ

N+1,1

(x) =



(x − x

_N

)/h

_N

, x ∈ [x

_N

, x

_N+1

) ,

0, otherwise.

The space of piecewise linear functions on Ω and its subspace consisting of functions vanishing at the endpoints of Ω are, respectively,

S

h

:= span {φ

1,1

, φ

_2,1

. . . , φ

_N+1,1

} and S

^◦h

:= span {φ

2,1

, . . . , φ

_N,1

}.

The spaces of bubble functions in Ω are defined by S

^kh

:= span {φ

1,k

, . . . , φ

N,k

} , where φ

_l,k

is an antiderivative of the Legendre polynomial P

_k−1

of degree k − 1 scaled to the subinterval [x

_l

, x

_l+1

]. More precisely, for l = 1, . . . , N and k = 2, 3, . . . , we define

φ

_l,k

(x) =

  √

2(2k − 1)/h

_l

 R

x

xl

P

_k−1

(y) dy , x ∈ [x

_l

, x

l+1

) ,

0, otherwise. (11)

2 The H

¹

-mixed finite element method C388

For p, q ∈ N and p, q > 2 , the finite dimensional subspaces of H

¹

(Ω) and H

¹₀

(Ω) are, respectively,

V

^q_h

:= S

h

∪ X

q k=2

S

^k_h

and

V

◦^p_h

:=

S

◦h

∪ X

p

k=2

S

^k_h

.

A semidiscrete approximation to (6)–(8) is to find (U, V) : [0, T ] → V

^◦p_h

× V

^q_h

such that for t ∈ (0, T ]:

h∂

_x

U(t), ∂

_x

χ

_h

i = hV(t), ∂

_x

χ

_h

i , for all χ

_h

∈ V

^◦p_h

, (12) h∂

_t

V(t), w

_h

i + ν h∂

_x

V(t), ∂

_x

w

_h

i = hUV(t), ∂

_x

w

_h

i , for all w

_h

∈ V

^qh

, (13) and

hV(0), w

_h

i = h∂

_x

u

0

, w

h

i for all w

h

∈ V

^q_h

. (14) Let the errors in the approximation of (6)–(8) by (12)–(14) be e(x, t) :=

u(x, t) − U(x, t) and f(x, t) := v(x, t) − V(x, t) . This leads to Proposition 3, the proof of which will be presented in a future paper.

Proposition 3. Assume that u, ∂

t

u ∈ L

^∞

(H

¹₀

(Ω) ∩ H

^p+1

(Ω)) and v, ∂

t

v ∈ L

^∞

(H

^q+1

(Ω)) . Assume further that U ∈ L

^∞

( V

^◦ph

) and V ∈ L

^∞

( V

^qh

) . Then, there exist positive constant C > 0 independent of h such that

ke(t)k

_j

6 Ch

min(p+1−j,q+1)

h

kuk

_L∞(Hp+1)

+ kvk

_L∞(H^q+1)

+ k∂

t

vk

_L2(H^q+1)

i , kf(t)k

_j

6 Ch

min(p+1,q+1−j)

h

kuk

_L∞(H^p+1)

+ kvk

_L∞(H^q+1)

+ k∂

_t

vk

_L2(H^q+1)

i . Now we show the computation of (U, V). With φ

_l,k

defined by (11), the solutions to (12)–(14) are

U(x, t) = X

N

l=2

U

_l,1

(t)φ

_l,1

(x) + X

N

l=1

X

p k=2

U

_l,k

(t)φ

_l,k

(x) , (15)

V(x, t) =

N+1

X

l=1

V

l,1

(t)φ

l,1

(x) + X

N

l=1

X

q k=2

V

l,k

(t)φ

l,k

(x) . (16)

2 The H

¹

-mixed finite element method C389

Let

α

^l,l_k,k⁰0

= hφ

l,k

, φ

l⁰,k⁰

i , α ¯

^l,l_k,k⁰0

= h∂

x

φ

l,k

, ∂

x

φ

l⁰,k⁰

i , β

^l,l_k,k⁰0

= hφ

l,k

, ∂

x

φ

l⁰,k⁰

i . (17) For each l = 1, . . . , N and r, r

⁰

= 2, 3, . . . we define a 2 × 2 matrix M

^l_1,1

, a 2 × (r − 1) matrix M

^l_1,r

, and an (r − 1) × (r

⁰

− 1) matrix M

^l_r,r0

with entries

[M

^l_1,1

]

ij

= α

l+j−1,l+i−1

1,1

, i, j = 1, 2 ,

[M

^l_1,r

]

ij

= α

^l,l+i−1_j,1

, i = 1, 2 , j = 2, . . . , r , [M

^l_r,r⁰

]

ij

= α

^l,l_j,i

, i = 2, . . . , r , j = 2, . . . , r

⁰

.

Similarly, we define matrices S

^l_1,1

, S

^l_1,r

, S

^l_r,r0

with ¯ α

^l,l_r,r⁰

, and B

^l_1,1

, B

^l_1,r

, B

^l_r,r0

with β

^l,l_r,r⁰

. We then define

M

^l_r

=

"

M

^l_1,1

M

^l_1,r

M

^l_1,r

T

M

^l_rr

#

, S

^l_r

=

"

S

^l_1,1

S

^l_1,r

S

^l_1,r

T

S

^l_rr

# ,

B

^l_r,r0

=

"

B

^l_1,1

B

^l_1,r0

B

^l_1,r

T

B

^l_r,r0

# .

The matrices M

^l_r

and S

^l_r

have size (r+1)×(r+1), whereas the matrix B

^l_r,r0

has size (r + 1) × (r

⁰

+ 1). The global matrices M

_r

, S

_r

and B

_r,r⁰

have elements M

^l_r

, S

^l_r

and B

^l_r,r0

, respectively. The sizes of M

r

and S

r

are (Nr + 1) × (Nr + 1) and the size of B

_r,r⁰

is (Nr + 1) × (Nr

⁰

+ 1).

For each l = 1, . . . , N we also define vectors

U

^l

= [U

_l,1

, U

_l+1,1

, U

_l,2

. . . , U

_l,p

]

^T

and V

^l

= [V

_l,1

, V

_l+1,1

, V

_l,2

. . . , V

_l,q

]

^T

, where the elements are defined in (15)–(16) and U

_1,1

and U

_N+1,1

are zero.

The vectors U and V are of size (Np + 1) × 1 and (Nq + 1) × 1, respectively,

and are assembled from the vectors U

^l

and V

^l

.

3 A posteriori error estimates and implementation issues C390

With the matrices defined above, the matrix representation of (12)–(13) is

S

p

U(t) = B

_p,q

V(t) , (18)

M

_q

∂

_t

V(t) + νS

_q

V(t) = G[U(t), V(t)] . (19) Here,

G(U, V) = [G

⁽⁰⁾

, G

⁽¹⁾

, . . . , G

^(N)

]

^T

is an (Nq + 1) × 1 vector with

G

⁽⁰⁾

= [hUV, φ

_1,1

i , hUV, φ

_2,1

i . . . , hUV, φ

_N+1,1

i]

^T

and

G

^(l)

= [hUV, φ

_l,2

i , hUV, φ

_l,3

i . . . , hUV, φ

_l,q

i]

^T

for l = 1, . . . , N . We use the Matlab ode solver to solve ( 18)–(19). The right hand side of (19) is computed by first solving (18) for a given V(t).

3 A posteriori error estimates and implementation issues

In this section we design methods to compute the error estimates. We infer that e and f satisfy

h∂

_x

e, ∂

_x

χ

_h

i = hf, ∂

_x

χ

_h

i for all χ

h

∈ V

^◦p_h

, (20) h∂

_t

f, w

h

i + ν h∂

_x

f, ∂

x

w

h

i − hef, ∂

_x

w

h

i − hUf, ∂

_x

w

h

i − heV, ∂

_x

w

h

i

= −ν h∂

_x

V, ∂

_x

w

_h

i + hUV, ∂

_x

w

_h

i − h∂

_t

V, w

_h

i for all w

_h

∈ V

^q_h

. (21) At t = 0 , from (8) and (14),

hf, w

_h

i = 0 for all w

_h

∈ V

^q_h

.

3 A posteriori error estimates and implementation issues C391

Due to (13), the right hand side of (21) vanishes. However, for the purpose of developing a posteriori error estimates, we keep these terms in the equation.

We approximate the exact errors e and f, respectively, by

E(x, t) = X

N

l=1

E

_l

(t)φ

_l,p+1

(x) ∈ S

^p+1_h

,

F(x, t) = X

N

l=1

F

_l

(t)φ

_l,q+1

(x) ∈ S

^q+1h

.

which are computed locally on each element (x

_l

, x

_l+1

), for l = 1, . . . , N , from the approximate solutions (U, V).

An accurate error estimation is one that satisfies

h

lim

→0

Θ(t) = 1 , t ∈ [0, T ] , (22) where

Θ(t) := E(t) ^

^ e(t) , with

^

e(t) := ke(t)k

_H1(Ω)

+ kf(t)k

_H1(Ω)

, E(t) := kE(t)k ^

_H1(Ω)

+ kF(t)k

_H1(Ω)

. We propose four different methods to compute E

_l

and F

_l

, l = 1, . . . , N . The first equation to be solved for each method is:

1. Nonlinear parabolic error estimate: (cf. (21))

h∂

_t

F

l

, φ

l,q+1

i

_l

+ ν h∂

x

F

l

, ∂

x

φ

l,q+1

i

_l

− hE

l

F

l

, ∂

x

φ

l,q+1

i

_l

− hUF

l

, ∂

x

φ

l,q+1

i

_l

− hVE

_l

, ∂

_x

φ

_l,q+1

i

_l

= −ν h∂

_x

V, ∂

_x

φ

_l,q+1

i

_l

+ hUV, ∂

_x

φ

_l,q+1

i

_l

− h∂

_t

V, φ

_l,q+1

i

_l

.

3 A posteriori error estimates and implementation issues C392

2. Nonlinear elliptic error estimate: We neglect the time rate of change in Method 1 so that

ν h∂

_x

F

_l

, ∂

_x

φ

_l,q+1

i

_l

− hE

_l

F

_l

, ∂

_x

φ

_l,q+1

i

_l

− hUF

_l

, ∂

_x

φ

_l,q+1

i

_l

− hVE

_l

, ∂

_x

φ

_l,q+1

i

_l

= −ν h∂

_x

V, ∂

_x

φ

_l,q+1

i

_l

+ hUV, ∂

_x

φ

_l,q+1

i

_l

− h∂

_t

V, φ

_l,q+1

i

_l

.

3. Linear parabolic error estimate: An additional reduction in the compu- tation cost is obtained by neglecting the nonlinear term hE

_l

F

_l

, ∂

_x

φ

_l,q+1

i

_l

in Method 1:

h∂

_t

F

l

, φ

l,q+1

i

_l

+ ν h∂

x

F

l

, ∂

x

φ

l,q+1

i

_l

− hUF

l

, ∂

x

φ

l,q+1

i

_l

− hVE

l

, ∂

x

φ

l,q+1

i

_l

= −ν h∂

_x

V, ∂

_x

φ

_l,q+1

i

_l

+ hUV, ∂

_x

φ

_l,q+1

i

_l

− h∂

_t

V, φ

_l,q+1

i

_l

. (23) 4. Linear elliptic error estimate: We neglect the nonlinear term in Method 2

so that

ν h∂

x

F

l

, ∂

x

φ

l,q+1

i

_l

− hUF

l

, ∂

x

φ

l,q+1

i

_l

− hVE

l

, ∂

x

φ

l,q+1

i

_l

= −ν h∂

x

V, ∂

x

φ

l,q+1

i

_l

+ hUV, ∂

x

φ

l,q+1

i

_l

− h∂

t

V, φ

l,q+1

i

_l

. Each equation in Methods 1–4 is coupled with (cf. (20))

h∂

_x

E

_l

, ∂

_x

φ

_l,p+1

i

_l

= hF

_l

, ∂

_x

φ

_l,p+1

i

_l

, (24) and an initial condition hF

_l

, φ

_l,q+1

i = h∂

_x

u

₀

, φ

_l,q+1

i − hV, φ

_l,q+1

i .

We finish this section with a discussion on implementation issues for the linear parabolic case. From (17), we have

h∂

_x

V, ∂

_x

φ

_l,q+1

i

_l

= V

_l+1,1

α ¯

^l+1,l_1,q+1

+ X

q k⁰=1

V

_l,k⁰

α ¯

^l,l_k0,q+1

:= T

₁

, and

h∂

_t

V, φ

_l,q+1

i

_l

= ∂

_t

V

_l+1,1

α

^l+1,l_1,q+1

+ X

q k⁰=1

∂

_t

V

_l,k⁰

α

^l,l_k0,q+1

:= T

₂

.

3 A posteriori error estimates and implementation issues C393

Also

α

^l,l_p+1,p+1

= h

_l

/((2p + 3)(2p − 1)) , α ¯

^l,l_p+1,p+1

= 2/h

_l

, and

β

^l,l_p+1,q+1

=

 



 

1/p(2q + 3)(2q + 1) , p = q + 1 ,

−1/p(2q + 1)(2q − 1) , p = q − 1 ,

0, otherwise.

By defining

β ¯

^l_¯k,k0,q

= φ

_l,¯k

φ

_l,k⁰

, ∂

_x

φ

_l,q+1

 , β ˜

^l_¯k,k0,q

= φ

_l+1,¯k

φ

_l,k⁰

, ∂

_x

φ

_l,q+1

 and

β ^

^l_¯k,k0,q

= φ

_l+1,¯k

φ

_l+1,k⁰

, ∂

_x

φ

_l,q+1

 , we have

hUF

_l

, ∂

x

φ

l,q+1

i

_l

= F

l

"

U

l+1,1

β ˜

^l_1,q+1,q

+ X

p k=1

U

l,k

β ¯

^l_k,q+1,q

#

:= T

3

F

l

,

hVE

_l

, ∂

x

φ

l,q+1

i

_l

= E

l

"

V

l+1,1

β ˜

^l_1,p+1,q

+ X

q k⁰=1

V

l,k⁰

β ¯

^l_k⁰_,p+1,q

#

:= T

4

E

l

, and

hUV, ∂

_x

φ

_l,q+1

i

_l

= U

_l+1,1

"

V

_l+1,1

β ^

^l_1,1,q

+ X

q k⁰=1

V

_l,k⁰

β ¯

^l_1,k0,q+1

#

+ X

p

k=1

U

l,k

"

V

l+1,1

β ^

^l_k,1,q

+ X

q k⁰=1

V

l,k⁰

β ¯

^l_k,k⁰_,q

#

:= T

5

.

The values of ¯ β

^l_k,k¯ 0,q

, ˜ β

^l_¯k,k0,q

and ^ β

^l_¯k,k0,q

can be computed using Maple.

By using the above definitions of T

₁

, . . . , T

₅

, (23) is rewritten as h

_l

(2q + 3)(2q − 1) ∂

_t

F

_l

(t) +  2ν h

_l

− T

₃



F

_l

(t) − T

₄

E

_l

(t) = −νT

₁

+ T

₅

− T

₂

.

4 Numerical results C394

Moreover, (24) is rewritten as

2E

_l

(t) = h

_l

β

^l,l_p+1,q+1

F

_l

(t) .

Then, by using the backward Euler formulation, we compute F

_l

(t

_j

) recursively using



d + 2ν

h

_l

− T

3

− T

4

β

^l,l_p+1,q+1

h

_l

2



F

l

(t

j

) = −νT

1

+ T

5

− T

2

+ dF

l

(t

j−1

) , (25) where d = h

_l

/(2q + 3)(2q − 1)(t

_j

− t

_j−1

) and t

_j

= j∆t for j = 1, 2, 3, . . . . The time step ∆t is chosen to be not less than h.

4 Numerical results

In this section, we present the numerical results obtained when solving (1)–(3) whose exact solutions are

u(x, t) = 2νπa sin(πx)

2 + a cos(πx) , v(x, t) = 2νπ

²

a cos(πx)

2 + a cos(πx) + 2ν[πa sin(πx)]

²

[2 + a cos(πx)]

²

, where a = exp(−π

²

νt) . The initial value is

u

₀

(x) = 2νπ sin(πx)/(2 + cos(πx)) .

In the following, we choose ν = 0.05 and p = q + 1 . The numerical results are also satisfactory for a larger ν, such as ν = 0.5 . We present the numerical results for ν = 0.05 only.

In the numerical experiment, we computed the approximate solution (U, V)

by solving (18)–(19). We then computed the errors e and f to check on the

convergence rate given by Proposition 3. Finally, we computed the error

estimations E and F by using the linear parabolic and linear elliptic a posteriori

error estimate methods 3 and 4 introduced in Section 3.

4 Numerical results C395

Table 1: The orders of convergence for (u, v) at t = 0.8 . p q N ke

h

(t)k

_H1(Ω)

κ

u

kf

_h

(t)k

_H1(Ω)

κ

v

2 1 20 1.1338E-3 6.6472E-2

40 2.8487E-4 1.993 3.3245E-2 0.999 80 7.1305E-5 1.998 1.6624E-2 1.000 160 1.7831E-5 1.999 8.3120E-3 1.000

3 2 20 2.2153E-5 2.8620E-3

40 2.7675E-6 3.000 7.1684E-4 1.997 80 3.4587E-7 3.000 1.7929E-4 1.999 100 1.7708E-7 3.000 1.1476E-4 1.999 Table 2: The effectivity indexes Θ at t = 0.8 .

Method 3 Method 4

p q h ^ e(t) E(t) ^ Θ(t) E(t) ^ Θ(t)

2 1 1/20 6.7606E-2 6.6984E-2 0.991 6.6543E-2 0.984 1/40 3.3530E-2 3.3364E-2 0.995 3.3308E-2 0.993 1/80 1.6695E-2 1.6652E-2 0.997 1.6645E-2 0.997 1/160 8.3298E-3 8.3188E-3 0.999 8.3180E-3 0.999 3 2 1/20 2.8841E-3 2.8736E-3 0.996 2.8671E-3 0.994 1/40 7.1961E-4 7.1833E-4 0.998 7.1792E-4 0.998 1/80 1.7964E-4 1.7948E-4 0.999 1.7946E-4 0.999 1/100 1.1493E-4 1.1485E-4 0.999 1.1484E-4 0.999

Table 1 presents the exact errors ke(t)k

_H1(Ω)

and kf(t)k

_H1(Ω)

for t = 0.8 . As

predicted by Proposition 3, the convergence rate is ke(t)k

_H1(Ω)

= O(h

^p

) and

kf(t)k

_H1(Ω)

= O(h

^p−1

) . Table 2 presents the computed a posteriori error

estimate ^ E and the effectivity index Θ(t), at t = 0.8 . For Method 3, when

solving (25) we chose ∆t = 0.4 . The results show that our a posteriori error

estimations are efficient.

5 Conclusion C396

5 Conclusion

We designed algorithms to estimate the true error when a model problem is solved using H

¹

-mfem. Our numerical experiments support our theoretical claims in Proposition 3 and (22). We emphasise that the computation of the error estimations (E

_l

, F

_l

) for l = 1, . . . , N can be carried out in parallel on each element (x

_l

, x

_l+1

). A theoretical study to show lim

_h→0

Θ(t) = 1 is the subject of a future paper.

References

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Author addresses

1. Sabarina Shafie, School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia.

mailto:[email protected]

References C398

2. Thanh Tran, School of Mathematics and Statistics, The University of New South Wales, Sydney 2052, Australia.

mailto:[email protected]