This Chapter aims to investigate the error estimation of numerical approximation to a class of semilinear parabolic problems. More specifically, the time discretization uses the backward Euler Galerkin method and the space discretization uses the finite element method for which the meshes are allowed to change in time. The key idea in our analysis is to adapt the elliptic reconstruction technique, introduced by Makridakis and Nochetto 2003, enabling us to use the a posteriori error estimators derived for elliptic models and to obtain optimal order in L∞H1 for Lipschitz and non-Lipschitz nonlinearities. In this Chapter, some challenges will be addressed to deal with nonlinear term by employing a continuation argument.
- A posteriori error estimates
- semilinear parabolic problems
- finite element approximation
- L∞ (H1) bounds in finite element approximation
- fully discrete semilinear parabolic approximation
The finite element method (FEM) consider is the most of flexibility common technique used for dealing with various kinds of application in many fields, for instance, in engineering, in chemistry and in biology. The derivation of a posteriori error estimates for linear and nonlinear parabolic problems are gaining increasing interest and there is a significant implementation of the method now are understandable and available in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9]. However, There is less progress has been made comparatively in the proving of a posteriori error bounds for semilinear parabolic problems [10, 11, 12, 13]. These estimations play a crucial rule in designing adaptive mesh refinement algorithms and consequently leading to a good accuracy while reducing the computational cost of the scheme.
The key technique used in the proofs is the elliptic reconstruction idea, introduced by Makridakis and Nochetto for spatially discrete conforming FEM  and extended to fully discrete conforming FEM by Lakkis and Makridakis  These ideas have been carried forward also to fully discrete schemes involving spatially non-conforming/dmethods in . The choice of this technique for deriving a posteriori error for parabolic problem is motivated by the following factors. First, elliptic reconstruction allows us to utilise the readily available elliptic a posteriori estimates  to bound the main part of the spatial error. Second, this technique combines the energy approach and appropriate pointwise representation of the error in order to arrive to optimal order a posteriori estimators in the -norm. As a result, this approach will lead to optimal order in both and -type norms, while the results obtained by the standard energy methods are only optimal order in -type norms.
The aim of this Chapter is to derive a posteriori error bounds for the fully discrete in two cases Lipschitz and non Lipschitz. Continuation Argument will be used to deal with nonlinear forcing terms.
Before we proceed with the error analysis, we require some auxiliary results that will be used in our analysis.
2.1 Functional spaces
Let is a function of time and space , we introduce the Bochner space where (is some real Banach space equipped with the norm which is the collection of all measurable functions : , more precisely, for any number
Lemma 1.1 (Continuous Gronwall inequality). Let for all and , then for almost every , reads
where . Furthermore, if and are non-negatives, gives
Proof: See .
Theorem 1.2 Given some , we have
Proof: See .
3. Model problem
Consider the semilinear parabolic problem as
where is a plane convex domain subset of with smooth boundary condition , where and . Let , and , , denote the standard Lebesgue and Hilbertian Sobolev spaces on a domain . For brevity, the norm of , , will be denoted by , and is induced by the standard -inner product, denoted by when , we shall use the abbreviations and .
for all . Here,
By using Cauchy-Schwarz inequality, the convercitivity and continuity of the bilnear form , viz.
with positive constants independent of .
4. Fully discrete backward Euler formulation
To introduce a backward Euler approximation of the time derivative paired with the standard conforming finite element method of the spatial operator. To this end, we will discretize the time interval into subintervals with and , and we denote by the local time step. We associate to each time-step a spatial mesh and the respective finite element space . The fully discrete scheme is defined as follows. Set to be a projection of onto some space subordinate to a mesh employed for the discretization of the initial condition. For , find such that the fully discrete, then reads as follows
where denotes the bilinear form defined on the mesh . Since , there exist , so that
where and are called the mass and stiffness matrices with element . We define the piecewise linear interpolant and time-dependent elliptic reconstruction as by the linear interpolant with respect to of the values and , viz.,
where denotes the linear Lagrange interpolation basis on the interval are defined as
We give here some essential definitions in the error analysis of the discrete parabolic equations.
projection operator ; The operator defined : such thatE15
for all .
Discrete elliptic operator: The elliptic operator defined : such that for , reads
Using the above projections, (10) can be expressed in distributional form as
5. Elliptic reconstruction
The aim of this section will be introduced the elliptic reconstruction operator and then discuss the related aposteriori error analysis for the backward Euler approximation. To do this, we define the elliptic reconstruction of as the solution of elliptic problem
for a given and . The crucial property, this operator is orthogonal with respect to such that
The following lemma is the elliptic reconstruction error bound in the and -norms To see the proof, we refer the reader to  for details.
Lemma 1.3 (Posteriori error estimates). For any , the following elliptic a posteriori bounds hold:
and defined in (18).
Lemma 1.4 (Main semilinear parabolic error equation). The following error bounds hold
Proof: To begin with, we first decompose the error as
By recalling (17), this becomes
Using elliptic reconstruction to split the error, gives
After using triangle inequality, the proof will be concluded.
The proof of the following Lemmas 1.5, 1.6, 1.7 in details, we refer to .
Lemma 1.5 (Temarol error estimate). Let be given by
Lemma 1.6 (Space-mesh error estimate). Let is defined by
Lemma 1.7 (Mesh change estimates). Let is given by
6. A posteriori error bound for fully discrete semilinear parabolic problems
The aim of this section is to study a posteriori error bound in -norm for nonlinear forcing terms. Both globally and locally Lipschitz continuous nonlinearities are considered.
6.1 A posteriori error analysis for the globally Lipschitz continuity case
Let us suppose that is defined on the whole of and satisfies globally Lipschitz continuous
where denotes the standard Euclidean norm on .
Lemma 1.8 (Data approximation error estimate). Suppose that the nonlinear reaction satisfying the globally Lipschitz continuous defined in (36), then, the following error bounds hold:
Proof: Using triangle inequality, written as
Applying Cauchy–Schwarz inequality and (36) along with Young’s inequality and Poincar’e-Friedrichs inequality, gives
The second term , reads
Finally, can be bounded by using Cauchy–Schwarz inequality, to obtain
Collecting all the results together, the proof will be finished.
Proof: Now, setting in 22, gives
Integrate the above from to then, we have
where defined in Lemmas 1.5, 1.6, 1.7 and 1.8, respectively. Summing up over : so that
Now, using Lemmas 1.5, 1.6, 1.7 and 1.8, reads
Selecting now be such that and using Gronwall’s inequality, imply
with . To finish the proof of lemma, we use a standard inequlty. For , .
and by taking
The proof already will be finished.
where defined in (20).
Proof: By decomposing into and , so that
and . Finally, the second term on the right hand side of (56) will be estimated via Lemma 1.9.
6.2 A posteriori error analysis for the locally Lipschitz continuity case
Let : is locally Lipschitz continuous for a.e. , in the sense that there exist real numbers and such that
Lemma 1.11 (Estimation of the nonlinear term). If the nonlinear reaction is satisfying the growth condition (58) with for , and with for , we have the bound
Proof: Applying triangle inequality, reads
can be bounded as follows
Now, we have
Applying the elementary inequality with and , so that , this becomes
Similarly, follows as
Collecting all these terms, we obtain
Using Holder’s inequality and Young’s inequality, we deduce that
Substituting this into our grand inequality yields
where and . From Gagliardo-Nirenberg inequality in Theorem 1.2, implies that
valid for all for and for . Combining this with the Poincar’e-Friedrichs inequality , yields
Putting all of the results together the proof will be finished.
where and are given in (20).
Proof: Now, setting in 22, and integrate from to along with summing up over : we have
Using Lemma 1.11, along with lemmas 1.3, 1.5, 1.6 and 1.7, imply
Upon observing that
Now combining two equations, we obtain
where . Since the left hand side of (78) depends continuously on , and our aim is to show that . To do this, assuming and , imply
and Grönwall inequality, thus, implies
Since and, suppose that the maximum size of the mesh is small enough that, for , satisfy
This leads to
Then, (81), becomes
This leads to contradictions, because of suppose to be .
The triangle inequality along with Lemma 1.3, imply that
By recalling (76), the proof already finished.
7. Adaptive algorithms
This section aims to explain an adaptive algorithm aiming to investigate the performance of the presented a posteriori bound from Theorems 1.10 and 1.12 for the backward-Euler method for the semilinear parabolic problem (6). To this end, the implementation of the adaptive algorithm will be based on the deal. II finite element library  to the present setting of semilinear problems. We shall write algorithm for Theorem 1.10. For the Theorem 1.12 will follow the same with some modifcations. To begin with, we have
The adaptive algorithm from , starts with an initial uniform mesh in space and with a given initial time step. Starting from a uniform square mesh of elements, the algorithm adapts the mesh to improve approximation to the initial condition using the initial condition estimator until some tolerance is satisfied. To adapt the timestep , the algorithm bisects a time interval not satisfying a user-defined temporal tolerance ttol, and leaves a time-interval unchanged if ttol.
Once the time-step is adapted, the algorithm performs spatial mesh refinement and coarsening, determined by the space indicator using the user-defined tolerances and , corresponding to refinement and coarsening, respectively. More specifically, we select the elements with the largest local contributions which result to for refinement. The spatial coarsening threshold is set to we select the elements with the smallest local contributions which result to for coarsening. The algorithm iterates for each time-step. We refer to  for the algorithm’s workflow and all implementation details. The following two algorithms give the backward Euler method to the system (12) and space-time adaptivity for Theorem 1.10.
Algorithm 1. The backward Euler method for solving the semilinear parabolic equation
1: Create a mesh with elements on the interval .
2: We disctize as , where is time step defined as .
4: for do
5: Calculate the mass and stiffness matrices and , and the load vector with entries
7: end for
Algorithm 2. Space-time adaptivity.
2: Pick .
3: Compute .
4: Compute from .
5: while or do bisction by refining all elements such that and coarsening all elements such that
6: if , then.
11: end if.
12: Compute .
13: Compute from .
14: end while
15: put .
16: while do
17: Calculute from .
18: while do
19: if then
24: end if
25: Compute from .
26: end while
27: Create from by refining all elements such that and coarsening all elements such that .
28: Compute from .
31: end while
The aim of this Chapter is to derive an optimal order a posteriori error estimates in term of the for the fully semilinear parabolic problems in two cases when Lipschitz and non Lipschitz are proved. The crucial tools in proving this error is the elliptic reconstruction techniques introduced by Makridakis and Nochetto 2003. This is consequently enabling us to use a posteriori error estimators derived for elliptic equation to obtain optimal order in terms of norm for Lipschitz and non-Lipschitz nonlinearities. Some challenges have to be overcome due to non-linearity on the forcing term depending on Gronwall’s Lemma and Sobolev embedding through continuation argument. Furthermore, this will give insight about designing adaptive algorithm, which allow use to control the cost of computations. In the future, this Chapter can be extended to the fully discrete case for semilinear parabolic interface problems in and norms [18, 20, 21, 22].
It is pleasure to thank Prof. E. Greogoulis (Department of Mathematics, University of Leicester, ), and Assistant Prof. A. Cangiani (Department of Mathematics, University of Nottingham, ) for their help and encouragement.