 # Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

In this work, we investigate the numerical approximation of the second order non-autonomous semilnear parabolic partial differential equation (PDE) using the finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the scheme toward the mild solution of the PDE under polynomial growth condition of the nonlinearity. Our convergence rate are obtain for smooth and non-smooth initial data and is similar to that of the autonomous case. Our convergence result for smooth initial data is very important in numerical analysis. For instance, it is one step forward in approximating non-autonomous stochastic partial differential equations by the finite element method. In addition, we provide realistic conditions on the nonlinearity, appropriated to achieve optimal convergence rate without logarithmic reduction by exploiting the smooth properties of the two parameters evolution operator.

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## 1 Introduction

Nonlinear partial differential equations are powerful tools in modelling real-world phenomena in many fields such as in geo-engineering. For instance processes such as oil and gas recovery from hydrocarbon reservoirs and mining heat from geothermal reservoirs can be modelled by nonlinear equations with possibly degeneracy appearing in the diffusion and transport terms. Since explicit solutions of many PDEs are rarely known, numerical approximations are forceful ingredients to quantify them. Approximations are usually done at two levels, namely space and time approximations. In this paper, we focus on spatial approximation of the following advection-diffusion problem with a nonlinear reaction term using the finite element method.

 ∂u∂t=A(t)u+F(t,u),u(0)=u0,t∈(0,T],T>0, (1)

on the Hilbert space , where is an open bounded subset of , with smooth boundary. The second order differential operator is given by

 A(t)u=d∑i,j=1∂∂xi(qij(t,x)∂u∂xj)−d∑j=1qj(t,x)∂u∂xj+q0(t,x)u, (2)

where and are smooth coefficients. Also, there exists , such that

 |qi,j(t,x)−qi,j(s,x)|≤c2|t−s|γ,x∈Λ,t,s∈[0,T],i,j∈{1,⋯,d}.

Moreover, satisfies the following ellipticity condition

 d∑i,j=1qij(t,x)ξiξj≥c|ξ|2,(t,x)∈[0,T]×¯¯¯¯Λ, (3)

where is a constant. The finite element approximation of (1) with constant linear operator are widely investigated in the scientific literature, see e.g. Stig2 ; Thomee2 ; Suzuki ; Antjd2 and the references therein. The finite volume method for was recently investigated in Tambueseul . If we turn our attention to the non-autonomous case, the list of references becomes remarkably short. In the linear homogeneous case (), the finite element approximation has been investigated in Luskin , (Suzuki, , Chapter III, Section 14.2). The linear inhomogeneous version of (1) () was investigated in Luskin ; Dimitri ; Thomee1 , (Suzuki, , Chapter III, Section 12) and the references therein. To the best of our knowledge, the nonlinear case is not yet investigated in the scientific literature. This paper fills that gap by investigating the error estimate of the finite element method of (1) with a nonlinear source , which is more challenging due to the presence of the unknown in the source term . This become more challenging when the nonlinear function satisfies the polynomial growth condition. Our strategy is based on an introduction of two parameters evolution operator by exploiting carefully its smooth regularity properties. Our key intermediate result, namely Lemma 3.1 generalizes (Thomee2, , Theorem 3.5) for time dependent and not necessary self-adjoint operators. It also generalizes (Thomee2, , Theorem 4.2), the results in (Suzuki, , Chapter III, Section 12) and in Luskin ; Dimitri ; Thomee1 to smooth and non-smooth initial data. Note that Lemma 3.1 for non-smooth initial data is of great important in numerical analysis. It is key to obtain the convergence of the finite element method for many nonlinear problems, including stochastic partial differential equations(SPDEs), see e.g. Kovcas ; Kruse1 ; Xiaojie2 and references therein for time independent SPDEs. In fact, in the case of SPDEs, due to the Itô-isometry formula or the Burkhölder Davis-Gundy inequality, the non-smooth version of Lemma 3.1 cannot be applied since it brings degenerates integrals, which causes difficulties in the error estimates or reduces considerably the order of convergence. Hence our result is more general than the existing results and also has many applications. The convergence rate achieved for semilinear problem is in agreement with many results in the literature on autonomous problems and on non-autonomous linear problems. More precisely, we achieve convergence order or , where is a regularity parameter defined in Assumption 2.1. Under optimal regularity of the nonlinear function or under a linear growth assumption on , we achieve optimal convergence order . Following Tambueseul and using the similar approach based on the two parameters evolution operator, this work can be extended to the finite volume method. The rest of this paper is structured as follows. In Section 2, the well-posedness results are provided along with the finite element approximation. The error estimate is analysed in Section 3 for both Lipschitz nonlinearity and polynomial growth nonlinearity.

## 2 Mathematical setting and numerical method

### 2.1 Notations, settings and well well-posedness problem

We denote by the norm associated to the inner product in the Hilbert space . We denote by the set of bounded linear operators in . Let be the set of continuous functions equipped with the norm , . Next, we make the following assumptions.

###### Assumption 2.1

The initial data belongs to , .

###### Assumption 2.2

The nonlinear function is Lipschitz continuous, i.e. there exists a constant such that

 ∥F(t,v)−F(s,w)∥≤K(|t−s|+∥v−w∥),s,t∈[0,T],v,w∈H. (4)

We introduce two spaces and , such that , depending on the boundary conditions of . For Dirichlet boundary conditions, we take . For Robin boundary condition, we take and

 H={v∈H2(Λ):∂v/∂vA+α0v=0,on∂Λ},α0∈R, (5)

where

stands for the differentiation along the outer conormal vector

. One can easily check that (Suzuki, , Chapter III, (11.14)) the bilinear operator , associated to defined by satisfies

 a(t)(v,v)≥λ0∥v∥21,v∈V,t∈[0,T], (6)

where is a positive constant, independent of . Note that is bounded in ((Suzuki, , Chapter III, (11.13))), so the following operator is well defined

 a(t)(u,v)=⟨−A(t)u,v⟩u,v∈V,t∈[0,T],

where is the dual space of V and the duality pairing between and . Identifying to its adjoint space , we get the following continuous and dense inclusions

 V⊂H⊂V∗,and therefore⟨u,v⟩H=⟨u,v⟩,u∈H,v∈V.

So if we want to replace by the scalar product of on , we therefore need to have , for . So the domain of is defined as

 D:=D(−A(t))=D(A(t))={u∈V,A(t)u∈H}.

It is well known that (Suzuki, , Chapter III, (11.11) & (11.11)) in the case of Dirichlet boundary conditions and in the case of Robin boundary conditions in (5). We write the restriction of to again which is therefore regarded as an operator of (more precisely the realization of ).

The coercivity property (6) implies that is a positive operator and its fractional powers are well defined (Stig2 ; Suzuki ). The following equivalence of norms holds Suzuki ; Stig2

 ∥v∥Hα(Λ) ≡ ∥((−A(t))α2v∥:=∥v∥α,v∈D((−A(t))α2)∩Hα(Λ),α∈[0,2]. (7)

It is well known that the family of operators generate a two parameters operators , see e.g. Sobolev or (Suzuki, , Page 832). The evolution equation (1) can be written as follows

 du(t)dt=A(t)u(t)+F(t,u(t)),u(0)=u0,t∈(0,T]. (8)

The following theorem provides the well posedness of problem (1) (or (8)).

###### Theorem 2.1

Sobolev Let Assumption 2.2 be fulfilled. If , then the initial value problem (1) has a unique mild solution given by

 u(t)=U(t,0)u0+∫t0U(t,s)F(s,u(s))ds,t∈(0,T]. (9)

Moreover, if Assumption 2.1 is fulfilled, then the following space regularity holds 111This estimate also holds when is replaced by its semi-discrete version defined in Section 2.2.

 ∥(−A(s))β2u(t)∥+∥F(u(t))∥≤C(1+∥(−A(s))β2u0∥),β∈[0,2),s,t∈[0,T]. (10)

### 2.2 Finite element discretization

Let be a triangulation of with maximal length . Let denotes the space of continuous and piecewise linear functions over the triangulation . We defined the projection from to by

 ⟨Phu,χ⟩H=⟨u,χ⟩H,χ∈Vh,u∈H. (11)

For any , the discrete operator is defined by

 ⟨Ah(t)ϕ,χ⟩H=⟨A(t)ϕ,χ⟩H=−a(t)(ϕ,χ),ϕ∈D∩Vh,χ∈Vh. (12)

The space semi-discrete version of problem (8) consists of finding such that

 duh(t)dt=Ah(t)uh(t)+PhF(t,uh(t)),uh(0)=Phu0,t∈(0,T]. (13)

For , we introduce the Ritz projection defined by

 ⟨−A(t)Rh(t)v,χ⟩H=⟨−A(t)v,χ⟩H=a(t)(v,χ),v∈V∩D,χ∈Vh. (14)

It is well known (see e.g. (Luskin, , (3.2)) or Suzuki ) that the following error estimate holds

 ∥Rh(t)v−v∥+h∥Rh(t)v−v∥H1(Λ)≤Chr∥v∥Hr(Λ),v∈V∩Hr(Λ),r∈[1,2]. (15)

The following error estimate also holds (see e.g. (Luskin, , (3.3)) or Suzuki )

 (16)

for any and , where and is the time derivative of . According to the generation theory, generates a two parameters evolution operator , see e.g. (Suzuki, , Page 839). Therefore the mild solution of (13) can be written as follows

 uh(t)=Uh(t,0)Phu0+∫t0Uh(t,s)PhF(s,uh(s))ds,t∈[0,T]. (17)

In the rest of this paper, stands for a constant indepemdent of , that may change from one place to another. It is well known (see e.g. (Suzuki, , Chapter III, (12.3) & (12.4))) that for any and , the following estimates hold222These estimates remain true if and are replaced by and respectively.

 ∥(−Ah(t))αUh(t,s)∥L(H)≤C(t−s)−α,∥Uh(t,s)(−Ah(s))α∥L(H)≤C(t−s)−α. (18)

## 3 Main result

### 3.1 Preliminaries result

We consider the following linear homogeneous problem: find such that

 w′=A(t)w,w(τ)=v,t∈(τ,T],% with0≤τ≤T. (19)

The corresponding semi-discrete problem in space is: find such that

 w′h(t)=Ah(t)wh,wh(τ)=Phv,t∈(τ,T],with0≤τ≤T. (20)

The following lemma will be useful in our convergence analysis.

###### Lemma 3.1

Let and . Let Assumption 2.2 be fulfilled. Then the following error estimate holds for the semi-discrete approximation (20)

 ∥w(t)−wh(t)∥=∥[U(t,τ)−Uh(t,τ)Ph]v∥≤Chr(t−τ)−(r−γ)2∥v∥γ,v∈D((−A(0))γ2).

Proof. We split the desired error as follows

 wh(t)−w(t)=(wh(t)−Rh(t)w(t))+(Rh(t)w(t)−w(t))≡θ(t)+ρ(t). (21)

Using the definition of and ((11)–(12)), we can prove exactly as in Stig2 that

 Ah(t)Rh(t)=PhA(t),t∈[0,T]. (22)

One can easily compute the following derivatives

 Dtθ = Ah(t)wh(t)−DtRh(t)w(t)−Rh(t)A(t)w(t), (23) Dtρ = DtRh(t)w(t)+Rh(t)A(t)w(t)−A(t)w(t). (24)

Endowing and the linear subspace with the norm , it follows from (15) that , . By the definition of the differential operator, it follows that for all . Hence for all and it follows from (24) that

 PhDtρ=DtRh(t)w(t)+Rh(t)A(t)w(t)−PhA(t)w(t). (25)

Adding and subtracting in (23) and using (22), it follows that

 Dtθ=Ah(t)θ−PhDtρ,t∈(τ,T], (26)

From (23), the mild solution of is given by

 θ(t)=Uh(t,τ)θ(τ)−∫tτUh(t,s)PhDsρ(s)ds. (27)

Splitting the integral part of (27) in two and integrating by parts the first one yields

 θ(t) = Uh(t,τ)θ(τ)+Uh(t,τ)Phρ(τ)−Uh(t,(t+τ)/2)Phρ((t+τ)/2) (28) + ∫(t+τ)/2τ∂∂s(Uh(t,s))Phρ(s)ds−∫t(t+τ)/2Uh(t,s)PhDsρ(s)ds.

Using the expression of , (see (21)) and the fact that , it holds that . Hence (28) reduces to

 θ(t)=−Uh(t,s)Phρ((t+τ)/2)+∫(t+τ)2τ∂∂s(Uh(t,s))Phρ(s)ds−∫t(t+τ)2Uh(t,s)PhDsρ(s)ds. (29)

Taking the norm in both sides of (29) and using (18) yields

 ∥θ(t)∥ ≤ C∥ρ((t+τ)/2)∥+∫(t+τ)2τ∥Uh(t,s)Ah(s)∥L(H)∥ρ(s)∥ds+∫t(t+τ)2∥Dsρ(s)∥ds (30) ≤ C∥ρ((t+τ)/2)∥+∫(t+τ)2τ(t−s)−1∥ρ(s)∥ds+∫t(t+τ)2∥Dsρ(s)∥ds.

Using (15) and (16), it holds that

 ∥ρ(s)∥≤Chr∥w(s)∥r,∥Dsρ(s)∥≤Chr(∥w(s)∥r+∥Dsw(s)∥r). (31)

Note that the solution of (19) can be represented as follows.

 w(s)=U(s,τ)v,s≥τ. (32)

Pre-multiplying both sides of (32) by and using (18) yields

 ∥∥(−A(s))r2w(s)∥∥ ≤ ∥∥∥(−A(s))r2U(s,τ)(−A(τ))−γ2∥∥∥L(H)∥∥∥(−A(τ))γ2v∥∥∥ (33) ≤ C(s−τ)−(r−γ)2∥∥∥(−A(τ))γ2v∥∥∥≤C(s−τ)−(r−γ)2∥v∥γ.

Therefore it holds that

 ∥w(s)∥r≤C(s−τ)−(r−γ)2∥v∥γ,0≤γ≤r≤2,τ

Substituting (34) in (31) yields

 ∥ρ(s)∥r≤Chr(s−τ)−(r−γ)2∥v∥γ. (35)

Taking the derivative with respect to in both sides of (32) yields

 Dsw(s)=−A(s)U(s,τ)v. (36)

As for (33), pre-multiplying both sides of (36) by and using (18) yields

 ∥Dsw(s)∥r≤C(s−τ)−1−(r−γ)2∥v∥γ. (37)

Substituting (34) and (37) in the second estimate of (31) yields

 ∥Dsρ(s)∥≤Chr((s−τ)−(r−γ)2∥v∥γ+(s−τ)−1−(r−γ)2∥v∥γ)≤Chr(s−τ)−1−(r−γ)2∥v∥γ. (38)

Substituting the first estimate of (31) and (38) in (30) and using (35) yields

 ∥θ(t)∥ ≤ Chr(t−τ)−(r−γ)2∥v∥γ+Chr∫t+τ2τ(t−s)−1(s−τ)−(r−γ)2∥v∥γds (39) + Chr∫tt+τ2(s−τ)−1−(r−γ)2∥v∥γds.

Using the estimate

 ∫t+τ2τ(t−s)−1(s−τ)−(r−γ)2ds+∫tt+τ2(s−τ)−1−(r−γ)2ds≤C(t−τ)−(r−γ)2,

it follows from (39) that

 ∥θ(t)∥≤Chr(t−τ)−(r−γ)2∥v∥γ. (40)

Substituting (40) and (35) in (21) completes the proof of Lemma 3.1.

### 3.2 Error estimate of the semilinear problem under global Lipschitz condition

###### Theorem 3.1

Let Assumptions 2.1 and 2.2 be fulfilled. Let and be defined by (9) and (17) respectively. Then the following error estimate holds

 ∥u(t)−uh(t)∥≤Ch2t−1+β/2+Ch2(1+ln(t/h2)),0

If in addition the nonlinearity satisfies the linear growth condition or if there exists small enough such that , , , then the following optimal error estimate holds

 ∥u(t)−uh(t)∥≤Ch2t−1+β/2,0

where is defined in Assumption 2.1.

###### Remark 3.1

Note that the hypothesis is not too restrictive. An example of class of nonlinearities for which such hypothesis is fulfilled is a class of functions satisfying , . Concrete examples are operators of the form , with continuous or bounded on .

###### Remark 3.2

It is possible to obtain an error estimate without irregularities terms of the form with a drawback that the convergence rate will not be , but will depend on the regularity of the initial data. The proof follows the same lines as that of Theorem 3.1 using Lemma 3.1 and this yields

 ∥u(t)−uh(t)∥≤Chβ,t∈[0,T].

Proof. of Theorem 3.1. We start with the proof of (41). Subtracting (17) form (9), taking the norm in both sides and using triangle inequality yields

 ∥u(t)−uh(t)∥ ≤ ∥U(t,0)u0−Uh(t,0)Phu0∥ (43) + ∥∥∥∫t0[U(t,s)F(s,u(s))−Uh(t,s)PhF(s,uh(s))]ds∥∥∥=:I0+I1.

Using Lemma 3.1 with and yields

 I0≤Ch2t−1+β/2∥u0∥β≤Ch2t−1+β/2. (44)

Using Assumption 2.2, (18) and (10) yields

 I1 ≤ ∫t0∥∥U(t,s)[F(s,u(s))−F(s,uh(s))]∥∥ds+∫t0∥∥[U(t,s)−Uh(t,s)Ph]F(s,uh(s))∥∥ds (45) ≤ C∫t0∥∥u(s)−uh(s)∥∥ds+C∫t0∥∥[U(t,s)−Uh(t,s)Ph]]F(s,uh(s))∥∥ds.

If , then using (18) easily yields . If , using Lemma 3.1 (with and ), and splitting the second integral in two parts yields

 I1 ≤ C∫t0∥u(s)−uh(s)∥ds+Ch2∫t−h20(t−s)−1ds+Ch2∫tt−h2(t−s)−1ds (46) ≤ C∫t0∥u(s)−uh(s)∥ds+Ch2(1+ln(t/h2)).

Substituting (46) and (44) in (43) and applying Gronwall’s lemma proves (41). To prove (55), we only need to re-estimate the term . Note that under assumption , using Lemma 3.1 (with and ) and (10), following the same lines as above one easily obtain . Let us now estimate under the hypothesis . Using Assumption 2.2, (10) and exploiting the mild solution (17) one easily obtain

 ∥F(t,uh(t))∥≤∥uh(t)∥≤C|t−s|ϵs−ϵ,∥F(s,uh(s))−F(t,uh(t))∥≤C|t−s|ϵs−ϵ, (47)

for some and any . Using Lemma 3.1 (with and ), triangle inequality and (47) yields

 I3 ≤ Ch2∫t0(t−s)−1∥∥F(s,uh(s))−F(t,uh(t))∥∥ds+Ch2∫t0(t−s)−1∥F(t,uh(t))∥ds ≤ Ch2∫t0(t−s)−1+ϵs−ϵds≤Ch2.

Hence the new estimate of is given below

 I1≤Ch2+C∫t0∥u(s)−uh(s)∥ds. (48)

Substituting (48) and (44) in (43) and applying Gronwall’s lemma proves (55) and the proof of Theorem 3.1 is completed.

### 3.3 Error estimate of the semilinear problem under polynomial growth condition

In this section, we take . We make the following assumptions on the nonlinearity.

###### Assumption 3.1

there exist two constants and such that the nonlinear function satisfies the following

 ∥F(w)∥ ≤ L1+L1∥w∥(1+∥w∥c1C),w∈H, (49) ∥F(w)−F(v)∥ ≤ L1∥w−v∥(1+∥u∥c1C+∥v∥c1C),w,v∈H. (50)

Let us recall the following Sobolev embedding (continuous embedding).

 D((−A(0))δ)⊂C(Λ,R),forδ>d2,d∈{1,2,3}. (51)

It is a classical solution that under Assumption 3.1 (8) has a unique mild solution satisfying333This remains true if is replaced by its discrete version . , see e.g. Sobolev . Hence using the Sobolev embbeding (51), it holds that

 ∥u(t)∥C≤C∥∥∥(−A(0))β2u(t)∥∥∥≤C,∥uh(t)∥C≤C∥∥∥(−A(0))β2uh(t)∥∥∥≤C,t∈[0,T]. (52)
###### Theorem 3.2

Let and be solution of (8) and (13) respectively. Let Assumptions 2.1 and 3.1 be fulfilled. Then the following error estimate holds

 ∥u(t)−uh(t)∥≤Ch2t−1+β/2+Ch2(1+ln(t/h2)),t∈[0,T]. (53)

If in addition there exists such that the nonlinearity satisfies the polynomial growth condition

 ∥F(t,v)∥≤C∥v∥c1∥v∥c2C, (54)

then the following optimal error estimate holds

 ∥u(t)−uh(t)∥≤Ch2t−1+β/2,0

Proof. The proof goes along the same lines as that of Theorem 3.1 by using appropriately Assumption 3.1 and (52).

###### Remark 3.3

It is possible in Theorem 3.2 to obtain convergence estimate without irregularities terms . But the convergence rate will depend on the regularity of the initial data and will be of the form

 ∥u(t)−uh(t)∥≤Chβ,t∈[0,T].
###### Remark 3.4

Assumption 3.1 is weaker than Assumption 2.2 and therefore include more nonlinearities. However, the price to pay when using Assumption 3.1 is that one requires more regularity on the initial data.

###### Remark 3.5

Let be polynomial of any order. The nonlinear operator is defined as the Nemytskii operator

 F(u)(x)=φ(u(x)),u∈H,x∈Λ,

is a concrete example satisfying Assumption 3.1.

In fact, let us assume without loss of generality that is polynomial of degree , that is

 φ(x)=l∑i=0aixi,x∈R. (56)

Note that the proofs in the cases are obvious. For any , using traingle inequality and the fact , , we obtain

 ∥F(v)∥2 = ∫Λ|F(v)(x)|2dx