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

By Younis Abid Sabawi

Submitted: June 18th 2020Reviewed: October 7th 2020Published: December 10th 2020

DOI: 10.5772/intechopen.94369

Downloaded: 15


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/dGmethods 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 L2H1and 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 ztxis a function of time tand space χ, we introduce the Bochner space LP0TXwhere (Xis 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,C1L10Tfor all T>0and zW1,1, then for almost every t0T, reads




where F0T=exp0TC1(ξt. Furthermore, if C0and C1are 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 Ωis a plane convex domain subset of Rk,ΩRkwith smooth boundary condition ∂Ω, where ut=u/t,T>0and fC1R. Let Lpω, 1pand 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 Ωand Ω.

Returning to the (6), multiplying by a test function vH01Ωand then integrate by parts, we arrive to (7) in weak form, which reads: find uL20TH01ΩH01(0,T,L2Ωfor 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,Ccoerpositive 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 0Tinto subintervals tn1tn,n=1,,Nwith t0=0and tN=T, and we denote by κn=tntn1the local time step. We associate to each time-step tNa spatial mesh Tnand the respective finite element space Vn;=VhpTn. The fully discrete scheme is defined as follows. Set Z0to be a projection of z0onto some space V0subordinate to a mesh T0employed for the discretization of the initial condition. For k=1,,n, find ZSnsuch that the fully discrete, then reads as follows


where Dn=Dtndenotes the cGbilinear 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=ΦjΦjand Ai,j=DΦjΦjare called the mass and stiffness matrices with element Fj,k=fΦjΦk. We define the piecewise linear interpolant Zand time-dependent elliptic reconstruction wtas by the linear interpolant with respect to tof the values Zn1and Zn, viz.,


where n1ndenotes the linear Lagrange interpolation basis on the interval Inare defined as


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

  1. L2projection operator Π0n; The operator defined Π0n: L2Vn,1nNsuch that


for all vL2Ω.

  • Discrete elliptic operator: The elliptic operator defined Ahn: H01ΩVnsuch 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 RbenH01Ωof Znas the solution of elliptic problem


    for a given vVnand gn=Π0nfnZnZnΠ0nZn1kn. The crucial property, this operator Rbenis orthogonal with respect to Dsuch that


    The following lemma is the elliptic reconstruction error bound in the H1and 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 gndefined 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,1,1nNbe given by






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


    we have




    Lemma 1.7 (Mesh change estimates). Let Tn,3,1nNis 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 fis 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 fsatisfying the globally Lipschitz continuous defined in (36), then, the following error bounds hold:




    Proof: Using triangle inequality, Tn,4written as


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


    The second term Ln,2, reads


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


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

    Lemma 1.9 Let zbe the exact solution of (7) and let Znbe 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=ρtin 22, gives


    Integrate the above from tn1to tnthen, we have


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


    By introducing




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


    Selecting now β>0be such that 2βCgCcoer>0and 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 zbe the exact solution of (7) and let Znbe its finite element approximation obtained by the backward Euler approximation (10). Then, for 1nN, the following a posteriori error bounds hold:


    where Φn,H12defined in (20).

    Proof: By decomposing Ztztinto ρand 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: RRis locally Lipschitz continuous for a.e. xtΩ0T, in the sense that there exist real numbers CL>0and γ0such that


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


    where N1t12CLtmax14γ,NZ121+4γZ2γand


    Proof: Applying triangle inequality, reads


    Jn,1can be bounded as follows


    Now, we have


    To estimate Z1,non 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+Cb2αwith Ca=zwand Cb=w, so that z2αCzw2α+Cw2α, this becomes


    Similarly, Z2,nfollows 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=12CL2tCmax116γand N22Z=121+4rZ2r. From Gagliardo-Nirenberg inequality in Theorem 1.2, implies that


    valid for all γ0for d=2and 0γ2for 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 zbe the exact solution of (7) and let Znbe its finite element approximation obtained by the backward Euler approximation (10). Then, for 1nN, the following a posteriori error bounds hold


    where Φn,L22and Φn,H12are given in (20).

    Proof: Now, setting v=ρtin 22, and integrate from tn1to tnalong with summing up over n=1: mwe 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>0and tm<T, imply


    and Grönwall inequality, thus, implies


    Since EtmZEtmZand, suppose that the maximum size hmaxof the mesh is small enough that, for h<hmax, satisfy


    This leads to


    Then, (81), becomes


    This leads to contradictions, because of tmsuppose 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 cGmethod 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 16×16elements, the algorithm adapts the mesh to improve approximation to the initial condition using the initial condition estimator Ψiniuntil 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 Ψspacejusing 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<stolfor 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 ODEsystem (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 nelements on the interval In.

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

    3: Settingα0=α0.

    4: for k=1,2,,ndo

    5: Calculate the mass and stiffness matrices Mand A, and the load vector Fwith 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 Z1from Z0.

    5: while Ψtime12>ttol+or maxΨspace12>stol+do bisction T0by refining all elements such that Ψspace12>stol+and coarsening all elements such that Ψspace12<stol

    6: if Ψtime12>ttol, then.

    7: n1n.

    8: Kn=Kn1,,κ2=κ1.

    9: κ2=κ12.

    10: κ1κ12.

    11: end if.

    12: Compute Z0.

    13: Compute Z1from Z0.

    14: end while

    15: put j=1, T1=T0, time=κ1.

    16: while time<Tdo

    17: Calculute Zjfrom Zj1.

    18: while Ψtimei2>ttoldo

    19: if Ψtime12>ttolthen

    20: n1n.

    21; κn=κn1,,κj+2=κj+1.

    22: κj+1=κj2.

    23; κjκj2.

    24: end if

    25: Compute Zjfrom Zj1.

    26: end while

    27: Create Tjfrom Tj1by refining all elements such that Ψspacei2>stol+and coarsening all elements such that Ψspacei2<stol.

    28: Compute Zjfrom 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 LH1for the fully semilinear parabolic problems in two cases when fuLipschitz 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 LH1norm 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+L2H1and LL2norms [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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    Younis Abid Sabawi (December 10th 2020). A Posteriori Error Analysis in Finite Element Approximation for Fully Discrete Semilinear Parabolic Problems [Online First], IntechOpen, DOI: 10.5772/intechopen.94369. Available from:

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