Optimal order a posteriori error estimates for fully discrete semilinear parabolic problems in $푳_\infty (푳_ퟐ )$ − norm are presented. Standard conforming Galerkin (continuous) finite element method is employed in space discretisation and backward Euler method is used in time discretisation. The main tools in deriving these error estimates are the elliptic reconstruction technique, energy arguments, with Grönwall's lemma and continuation argument. These error bounds can be used in designing efficient adaptive finite element schemes which lead to significant savings in the computational costs such as implementation time, memory and data size.

1. A Posteriori Error Analysis for Fully Discrete Semilinear
Parabolic Problems
Mohammad Sabawi
Department of Mathematics, College of Education for Women,
Tikrit University,
Tikrit, Salah Al-Dean Province, Iraq,
Email: mohammad.sabawi@tu.edu.iq
Research No. : 1570494824

2. Outline
•Introduction and Literature Review
•The Mathematical Model and Settings
•Error Analysis for Fully Discrete Case
1. Globally Lipschitz Continuous Source Term
2. Locally Lipschitz Continuous Source Term

4. Literature Review
• Makridakis and Nochetto (2003), derived a posteriori bounds for the
semidiscrete case for linear parabolic problems by introducing elliptic
reconstruction technique.
• Makridakis and Nochetto (2006), introduced a novel time reconstruction
operator and used it in deriving a posteriori error bounds for the
semidiscrete linear and nonlinear evolutionary problems.
• Schötzau and Wihler (2010), obtained a posteriori error estimates for ℎ𝑝-
version DG timestepping by using time reconstruction.
• Georgoulis, Lakkis and Wihler (2018), presented a new a posteriori error
bounds for the fully discrete linear parabolic problems using a novel space-
time reconstruction.
• Cangiani, Georgoulis and Sabawi (2018), obtained optimal order a posteriori
error bounds in the 𝐿$ 𝐿% − and 𝐿%(𝐻)) − norms for fully-discrete space-
time implicit-explicit (IMEX) ℎ𝑝-version discontinuous Galerkin timestepping
scheme for semilinear parabolic problems.
• Sabawi (2018), derived optimal order a posteriori error bounds for
semidiscrete semilinear parabolic problems.

5. The Mathematical Model and Settings
We consider as a mathematical model the following initial-boundary value
problem
𝑢, − Δ𝑢 = 𝑓 𝑢 𝑖𝑛 Ω×𝐼,
𝑢 . , 0 = 𝑢8 . 𝑖𝑛 Ω, (1)
𝑢 = 0 𝑜𝑛 𝜕Ω×I,
for some known function u8 ∈ L% Ω , u = u t, x , x ∈ ℝA
d ≥ 1
where Ω is a bounded domain in ℝA
and 𝜕Ω is the boundary of Ω, I =
0, 𝑇 and f: I×ℝA
×ℝ ⟶ ℝ is a Lipschitz continuous function.
The weak form of (1) reads: find u ∈ 𝐻8
)
Ω such that
𝑢,, 𝑣 + 𝑎 𝑢, 𝑣 = 𝑓 𝑢 , 𝑣 , ∀𝑣 ∈ 𝐻8
)
Ω , 𝑡 > 0. (2)

6. Elliptic Reconstruction
We define the elliptic reconstruction 𝑈: = 𝑅𝑢R 𝜖 𝐻8
)
of the approximate solution 𝑢R by:
𝑎 𝑈 𝑡 , 𝜒 = 𝑎 𝑅𝑢R 𝑡 , 𝜒 = 𝐴V
𝑢R 𝑡 , 𝜒 ∀𝜒 𝜖 𝑉V
, 𝑡 𝜖 𝐼V, (3)
where 𝑉V
: = X𝑉V
∩ 𝐻8
)
, X𝑉V
: = 𝑣 ∈ 𝐻)
: ∀ 𝐾 ∈ 𝑇V ∶ 𝑣|] ∈ 𝑃_
, 𝐼V = 𝑡V`), 𝑡V , 𝐾 is an
element in the triangulation 𝑇V of the computational domain Ω, and 𝑃_
is the space of
polynomials of degree less than or equal to 𝑚.
𝐴V: 𝑉V → 𝑉V is the discrete elliptic operator defined by
𝐴V
𝑣, 𝜒 = 𝑎 𝑣, 𝜒 ∀𝜒, 𝑣 𝜖 𝑉V
, (4)
and 𝑅: 𝑉V → 𝐻8
)
is the elliptic reconstruction operator. From (3) and (4), we obtain

7. 𝑎 𝑈 𝑡 , 𝜒 = 𝐴V
𝑢R 𝑡 , 𝜒 = 𝑎 𝑢R 𝑡 , 𝜒 ∀𝜒 𝜖 𝑉V
, 𝑡 𝜖 𝐼V. (5)
Hence
𝑢R = 𝑃R
V
𝑈, (6)
where 𝑃R
V
is the elliptic projection operator. This means that U is the approximate
solution of the elliptic problem its exact solution is the elliptic reconstruction
function U.
The backward Euler-Galerkin finite element discretisation of (2) with finite
element space Ve
leads to the following fully discrete scheme
∀ 𝑛 ∈ 1, 𝑁 find 𝑢R
V
∈ 𝑉V
such that
𝜏V
`)
𝑢R
V
− 𝑢R
V`)
, 𝜃V
+ 𝑎 𝑢R
V
, 𝜃V
= 𝑓 𝑢R
V
, 𝜃V
, ∀ 𝜃V
∈ 𝑉V
,
𝑢R
8
= Π8
𝑢8
, (7)
where Π8
is suitable projection or interpolation operator and 𝜏V = 𝑡V − 𝑡V`) .

8. Splitting Error Formula
The error 𝑒 ∶= 𝑢 − 𝑢R can be split in the following way:
𝑒 ∶= 𝑢 − 𝑈 + 𝑈 − 𝑢R = 𝜌 + 𝜖, (11)
where 𝜌: = 𝑢 − 𝑈 and 𝜖: = 𝑈 − 𝑢R and 𝑈 is the elliptic
reconstruction of the approximate spatially discrete solution 𝑢R,
𝑈 ∶= 𝑅𝑢R, where 𝑅 is the elliptic reconstruction operator.

9. Elliptic A Posteriori Error Bound
If 𝑢R is the approximate solution of the exact solution of
(1) and 𝑈 its elliptic reconstruction, then we have the
following a posteriori error bound:
𝜖 = ∥ 𝑈 − 𝑢R ∥ ≤ 𝐹 𝑢R, 𝑓 𝑢R , (8)
and this error can be bounded by optimal order a
posteriori quantities.

10. Remarks:
• Note that the elliptic part of the error 𝜖 is computable since it
depends upon the approximate solution 𝑢R and the data of the
problem, and it is available in the literature.
•The parabolic part of the error 𝜌 is not computable since it
depends on the exact solution 𝑢 which in most cases
unavailable. Also, it depends on the theoretical quantity 𝑈.
• The parabolic part of the error 𝜌 satisfies a perturbation of the
original PDE whose right-hand side can be bounded a posteriori
in an optimal way.

11. The main Goals
•Bounding the parabolic part of the error 𝜌 by a quantity
depends on the elliptic a posteriori error 𝜖
𝜌 = 𝑢 − 𝑈 ≤ 𝐸 𝜖 = 𝐸(𝑢R, 𝑓(𝑢R)). (9)
•Then, bounding the total error by quantities depend only on
elliptic part of the error 𝜖
𝑒 ≤ 𝜌 + 𝜖 ≤ 𝐸(𝑢R, 𝑓(𝑢R)) + 𝐹(𝑢R, 𝑓(𝑢R)). (10)

12. A POSTERIORI ERROR ANALYSIS FOR FULLY DISCRETE SEMILINEAR
PARABOLIC EQUATIONS
The function 𝑓 is globally Lipschitz continuous if it satisfies
𝑓 𝑣 − 𝑓 𝑤 ≤ 𝐶 𝑣 − 𝑤 , (12)
where 𝐶 is an arbitrary real constant, and it is not necessarily the same at each occurrence.
Theorem 1 (𝐿$ 𝐿% Error Bound for Globally Lipschitz Case). If the right-hand side
function 𝑓 is globally Lipschitz continuous, then the following conditional estimate
holds
𝑢 − 𝑢R tu 8,,;tw
≤ 𝐸) 𝑡, 𝜖, 𝑢R 𝑒xy
z
w + 𝜖 tu 8,,;tw
, (13)
𝐸) = 𝐸) 𝑡, 𝜖, 𝑢R = ∫8
,
| ϵ~
% + 𝐶 ϵ 2 + 𝑙V`)
%
𝑡 𝐴V`) 𝑢R
V`)
− 𝐴V 𝑢R
V %
+ 𝑓V − 𝑓 𝑢R
%
€+ 𝑃8
V
𝑓V − 𝑓V + 𝜏V
`) 𝑃8
V
𝑢R
V`)
− 𝑢R
V`) %
𝑑𝜏,
where 𝑃8
V
is the 𝐿% projection operator.

13. A Posteriori Error Analysis for the Locally Lipschitz
Continuity Case
It said to be that 𝑓 satisfies the following growth condition if
𝑓 𝑣 − 𝑓 𝑤 ≤ 𝐶 𝑣 − 𝑤 1 + 𝑣 + 𝑤 ‚
, 𝛼 > 0. (14)
Lemma 1 (Bounding the Nonlinear Term). If the nonlinear source
function satisfies the growth condition (14), with 0 ≤ α < 2 for d =
2, and with 0 ≤ α ≤ 4/3 for d = 3. Then, we have the bound
𝑓 𝑢 − 𝑓 𝑢R , 𝜌 ≤ 𝐶 Š𝐶 𝜖 ‚
𝜖 ‹
%
+ 𝐺 𝑢R 𝜖 %
+ 𝐶 Š𝐶 𝜌 ‚
𝜌 ‹
%
+ 𝐺 𝑢R 𝜌 %
, (15)
where 𝐺 𝑢R = 1 + 𝑢R tu •
‚
and Š𝐶 is a real constant and also
called Sobolev imbedding constant.

14. The Error Bound for The Locally Lipschitz Case
Theorem 2 (𝑳$ 𝑳 𝟐 Error Bound for Locally Lipschitz Case). Assume the
hypotheses of Lemma 1, the following conditional estimate holds
𝑢 − 𝑢R tu 8,,;tw
≤ 𝜖 tu 8,,;tw
+
𝐹 𝑡, 𝑢R 𝛿 + 𝐶 Š𝐶 4𝛿𝐹 𝑡, 𝑢R
‘’w
w , (16)
where
𝛿 = “
8
,
ϵ~
% + CG uh ϵ 2 + CXC 𝜖 ‚ 𝜖 ‹
% 𝑑𝜏 +
𝑙V`)
%
𝑡 𝐴V`)
𝑢R
V`)
− 𝐴V
𝑢R
V %
+ 𝑓V
− 𝑓 𝑢R
%
+ 𝑃8
V
𝑓V
− 𝑓V
+ 𝜏V
`)
𝑃8
V
𝑢R
V`)
− 𝑢R
V`) %
,
and
𝐹 𝑡, 𝑢R = 𝑒
xz ∫—
˜
™ š› œ•
.

15. Thank You
for Your Attention.