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A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear Parabolic Problems

Written By

Younis Abid Sabawi

Submitted: 18 June 2020 Reviewed: 07 October 2020 Published: 10 December 2020

DOI: 10.5772/intechopen.94369

From the Edited Volume

Finite Element Methods and Their Applications

Edited by Mahboub Baccouch

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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

1. Introduction

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 [2] and extended to fully discrete conforming FEM by Lakkis and Makridakis [3] These ideas have been carried forward also to fully discrete schemes involving spatially non-conforming/dG methods in [14]. 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 [2] 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 LL2-norm. As a result, this approach will lead to optimal order in both L2H1 and LL2-type norms, while the results obtained by the standard energy methods are only optimal order in L2H1-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.


2. Preliminaries

Before we proceed with the error analysis, we require some auxiliary results that will be used in our analysis.

2.1 Functional spaces

Let ztx is a function of time t and space χ, we introduce the Bochner space LP0TX where (X is some real Banach space equipped with the norm X) which is the collection of all measurable functions v: 0TX, more precisely, for any number r1


such that


Lemma 1.1 (Continuous Gronwall inequality). Let C0,C1L10T for all T>0 and zW1,1, then for almost every t0T, reads




where F0T=exp0TC1(ξt. Furthermore, if C0 and C1 are non-negatives, gives


Proof: See [15].

Theorem 1.2 Given some p2, we have


Proof: See [16].


3. Model problem

Consider the semilinear parabolic problem as


where Ω is a plane convex domain subset of Rk,ΩRk with smooth boundary condition ∂Ω, where ut=u/t,T>0 and fC1R. Let Lpω, 1p and Hrω, rR, denote the standard Lebesgue and Hilbertian Sobolev spaces on a domain ωΩ. For brevity, the norm of L2ωH0ω, ωΩ, will be denoted by ω, and is induced by the standard L2ω-inner product, denoted by ω; when ω=Ω, we shall use the abbreviations Ω and Ω.

Returning to the (6), multiplying by a test function vH01Ω and then integrate by parts, we arrive to (7) in weak form, which reads: find uL20TH01ΩH01(0,T,L2Ω for almost every t0T, this becomes


for all vH01Ω. Here,


By using Cauchy-Schwarz inequality, the convercitivity and continuity of the bilnear form D, viz.


with Ccont,Ccoer positive constants independent of w, v.


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 0T into subintervals tn1tn,n=1,,N with t0=0 and tN=T, and we denote by κn=tntn1 the local time step. We associate to each time-step tN a spatial mesh Tn and the respective finite element space Vn;=VhpTn. The fully discrete scheme is defined as follows. Set Z0 to be a projection of z0 onto some space V0 subordinate to a mesh T0 employed for the discretization of the initial condition. For k=1,,n, find ZSn such that the fully discrete, then reads as follows


where Dn=Dtn denotes the cG bilinear form defined on the mesh Tn. Since ZnVn, there exist αitR,j=0,1,2,,Nh, so that


is the basis functions. After plugging (11) into (10), yields a nonlinear system of ordinary differential equations


where Mi,j=ΦjΦj and Ai,j=DΦjΦj are called the mass and stiffness matrices with element Fj,k=fΦjΦk. We define the piecewise linear interpolant Z and time-dependent elliptic reconstruction wt as by the linear interpolant with respect to t of the values Zn1 and Zn, viz.,


where n1n denotes the linear Lagrange interpolation basis on the interval In are defined as


We give here some essential definitions in the error analysis of the discrete parabolic equations.

  1. L2 projection operator Π0n; The operator defined Π0n: L2Vn,1nN such that


    for all vL2Ω.

  2. Discrete elliptic operator: The elliptic operator defined Ahn: H01ΩVn such that for vH01Ω, 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 RbenH01Ω of Zn as the solution of elliptic problem


for a given vVn and gn=Π0nfnZnZnΠ0nZn1kn. The crucial property, this operator Rben is orthogonal with respect to D such that


The following lemma is the elliptic reconstruction error bound in the H1 and L2-norms To see the proof, we refer the reader to [3] for details.

Lemma 1.3 (Posteriori error estimates). For any ZnVn, the following elliptic a posteriori bounds hold:




and gn 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


where Zt=Zn1Znκn. Subtracting (24) from (7), gives


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 [3].

Lemma 1.5 (Temarol error estimate). Let Tn,1,1nN be given by






Lemma 1.6 (Space-mesh error estimate). Let Tn,2,1nN is defined by


we have




Lemma 1.7 (Mesh change estimates). Let Tn,3,1nN is given by


such that




6. A posteriori error bound for fully discrete semilinear parabolic problems

The aim of this section is to study a posteriori error bound in LH1-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 f is defined on the whole of and satisfies globally Lipschitz continuous


where denotes the standard Euclidean norm on R1.

Lemma 1.8 (Data approximation error estimate). Suppose that the nonlinear reaction f satisfying the globally Lipschitz continuous defined in (36), then, the following error bounds hold:




Proof: Using triangle inequality, Tn,4 written as


Applying Cauchy–Schwarz inequality and (36) along with Young’s inequality and Poincar’e-Friedrichs inequality, Ln,1 gives


The second term Ln,2, reads


Finally, Ln,3 can be bounded by using Cauchy–Schwarz inequality, to obtain


Collecting all the results together, the proof will be finished.

Lemma 1.9 Let z be the exact solution of (7) and let Zn be its finite element approximation obtained by the backward Euler approximation (10). Then, for 1nN, the following a posteriori error bounds hold:




Proof: Now, setting ϕ=ρt in 22, gives


Integrate the above from tn1 to tn then, we have


where Tn,i,i=1,2,3,4 defined in Lemmas 1.5, 1.6, 1.7 and 1.8, respectively. Summing up over n=1: m so that


By introducing




Now, using Lemmas 1.5, 1.6, 1.7 and 1.8, reads


Selecting now β>0 be such that 2βCgCcoer>0 and using Gronwall’s inequality, imply


with EGm1n=1m2Cgβκnexp2CgβΣn<j<mkj. To finish the proof of lemma, we use a standard inequlty. For a0a1an, b0b1bnRm+1.




and by taking


The proof already will be finished.

Theorem 1.10 Let z be the exact solution of (7) and let Zn be its finite element approximation obtained by the backward Euler approximation (10). Then, for 1nN, the following a posteriori error bounds hold:


where Φn,H12 defined in (20).

Proof: By decomposing Ztzt into ρ and ε, so that


To be able to bound the first term on the right hand side of (56), using (13), this becomes


and ρ02=w0z022ε02+2z0Z02. 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 f: RR is locally Lipschitz continuous for a.e. xtΩ0T, in the sense that there exist real numbers CL>0 and γ0 such that


Lemma 1.11 (Estimation of the nonlinear term). If the nonlinear reaction f is satisfying the growth condition (58) with 0r<2 for d=2, and with 0r4/3 for d=3, we have the bound


where N1t12CLtmax14γ,NZ121+4γZ2γ and


Proof: Applying triangle inequality, reads


Jn,1 can be bounded as follows


Now, we have


To estimate Z1,n on the first term in the right hand side of (62), we use the Cauchy–Schwarz inequality and (58) to obtain


Applying the elementary inequality Ca+Cb2αCCa2α+Cb2α with Ca=zw and Cb=w, so that z2αCzw2α+Cw2α, this becomes


Similarly, Z2,n 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 N12t=12CL2tCmax116γ and N22Z=121+4rZ2r. From Gagliardo-Nirenberg inequality in Theorem 1.2, implies that


valid for all γ0 for d=2 and 0γ2 for d=3. Combining this with the Poincar’e-Friedrichs inequality ρCρ, yields




Putting all of the results together the proof will be finished.

Theorem 1.12 Let z be the exact solution of (7) and let Zn be its finite element approximation obtained by the backward Euler approximation (10). Then, for 1nN, the following a posteriori error bounds hold


where Φn,L22 and Φn,H12 are given in (20).

Proof: Now, setting v=ρt in 22, and integrate from tn1 to tn along with summing up over n=1: m 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


To bound of the nonlinear term of above equation, we shall employ a continuation argument in the spirit of [17, 18]. To do that, we consider the set


where EtmZ=exp0tmN12tN22Zdt. Since the left hand side of (78) depends continuously on t, and our aim is to show that Mn=0T. To do this, assuming tm=maxMn>0 and tm<T, imply


and Grönwall inequality, thus, implies


Since EtmZEtmZ and, suppose that the maximum size hmax of the mesh is small enough that, for h<hmax, satisfy


This leads to


Then, (81), becomes


This leads to contradictions, because of tm suppose to be tm=maxMn.

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 cG 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 [19] 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 [15], starts with an initial uniform mesh in space and with a given initial time step. Starting from a uniform square mesh of 16×16 elements, the algorithm adapts the mesh to improve approximation to the initial condition using the initial condition estimator Ψini until some tolerance is satisfied. To adapt the timestep κj, the algorithm bisects a time interval not satisfying a user-defined temporal tolerance Ψtimejttol, and leaves a time-interval unchanged if ϒtimejttol.

Once the time-step is adapted, the algorithm performs spatial mesh refinement and coarsening, determined by the space indicator Ψspacej using the user-defined tolerances stol+ and stol, corresponding to refinement and coarsening, respectively. More specifically, we select the elements with the largest local contributions which result to Ψspacej>stol+ for refinement. The spatial coarsening threshold is set to stol=0.001stol+; we select the elements with the smallest local contributions which result to Ψspacej<stol for coarsening. The algorithm iterates for each time-step. We refer to [15] for the algorithm’s workflow and all implementation details. The following two algorithms give the backward Euler method to the ODE 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 n elements on the interval In.

2: We disctize In as 0=t1<t2<t3,,<tn=T, where n is time step defined as κn=tntn1.

3: Settingα0=α0.

4: fork=1,2,,ndo

5: Calculate the mass and stiffness matrices M and A, and the load vector F with entries


6: Solve


7: end for

Algorithm 2. Space-time adaptivity.

1: Input a,b,f,z0,T,Ω,n,T,ttol,stol+,stol

2: Pick κ1,,κn=Tn.

3: Compute Z0.

4: Compute Z1 from Z0.

5: whileΨtime12>ttol+ormaxΨspace12>stol+ do bisction T0 by refining all elements such that Ψspace12>stol+ and coarsening all elements such that Ψspace12<stol

6: ifΨtime12>ttol, then.

7: n1n.

8: Kn=Kn1,,κ2=κ1.

9: κ2=κ12.

10: κ1κ12.

11: end if.

12: Compute Z0.

13: Compute Z1 from Z0.

14: end while

15: put j=1, T1=T0, time=κ1.

16: whiletime<T do

17: Calculute Zj from Zj1.

18: whileΨtimei2>ttoldo

19: ifΨtime12>ttolthen

20: n1n.

21; κn=κn1,,κj+2=κj+1.

22: κj+1=κj2.

23; κjκj2.

24: end if

25: Compute Zj from Zj1.

26: end while

27: Create Tj from Tj1 by refining all elements such that Ψspacei2>stol+ and coarsening all elements such that Ψspacei2<stol.

28: Compute Zj from Zj1.

29: timetime+κj.

30: j1j.

31: end while


8. Conclusion

The aim of this Chapter is to derive an optimal order a posteriori error estimates in term of the LH1 for the fully semilinear parabolic problems in two cases when fu 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 LH1 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 LL2+L2H1 and LL2 norms [18, 20, 21, 22].

Notes/thanks/other declarations

It is pleasure to thank Prof. E. Greogoulis (Department of Mathematics, University of Leicester, UK), and Assistant Prof. A. Cangiani (Department of Mathematics, University of Nottingham, UK) for their help and encouragement.


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Written By

Younis Abid Sabawi

Submitted: 18 June 2020 Reviewed: 07 October 2020 Published: 10 December 2020