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.


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) = u 0 , t ∈ (0, T ], T > 0, (1) on the Hilbert space H = L 2 (Λ), where Λ is an open bounded subset of R d (d = 1, 2, 3), with smooth boundary.The second order differential operator A(t) is given by q j (t, x) ∂u ∂x j + q 0 (t, x)u, where q i,j , q j and q 0 are smooth coefficients.Also, there exists c 1 ≥ 0, 0 < γ ≤ 1 such that |q i,j (t, x) − q i,j (s, x)| ≤ c 2 |t − s| γ , x ∈ Λ, t, s ∈ [0, T ], i, j ∈ {1, • • • , d}.
Moreover, q i,j satisfies the following ellipticity condition where c > 0 is a constant.The finite element approximation of (1) with constant linear operator A(t) = A are widely investigated in the scientific literature, see e.g.[4,11,1,8] and the references therein.The finite volume method for A(t) = A was recently investigated in [10].If we turn our attention to the non-autonomous case, the list of references becomes remarkably short.In the linear homogeneous case (F (t, u) = 0), the finite element approximation has been investigated in [6], [1, Chapter III, Section 14.2].
The linear inhomogeneous version of (1) (F (t, u) = f (t)) was investigated in [6,5,7], [1, 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 F (t, u), which is more challenging due to the presence of the unknown u in the source term F .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.F or under a linear growth assumption on F , we achieve optimal convergence order O(h 2 t −1+β/2 ).Following [10] 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.

Notations, settings and well well-posedness problem
We denote by • the norm associated to the inner product We introduce two spaces H and V , such that H ⊂ V , depending on the boundary conditions of −A(t).For Dirichlet boundary conditions, we take V = H = H 1 0 (Λ).For Robin boundary condition, we take V = H 1 (Λ) and where ∂v/∂v A stands for the differentiation along the outer conormal vector v A .One can easily check that [1, Chapter III, (11.14 ′ )] the bilinear operator a(t), associated to −A(t) where λ 0 is a positive constant, independent of t.Note that a(t Chapter III, (11.13)]), so the following operator where V * is the dual space of V and •, • the duality pairing between V * and V .Identifying H to its adjoint space H * , we get the following continuous and dense inclusions So if we want to replace •, • by the scalar product of •, • H on H, we therefore need to have A(t)u ∈ H, for u ∈ V .So the domain of −A(t) is defined as It is well known that [1, Chapter III, (11.11) & (11.11 ′ )] in the case of Dirichlet boundary and in the case of Robin boundary conditions D = H in (5).We write the restriction of A(t) : V −→ V * to D (A(t)) again A(t) which is therefore regarded as an operator of H (more precisely the H realization of A(t)).
The coercivity property (6) implies that −A(t) is a positive operator and its fractional powers are well defined ( [4,1]).The following equivalence of norms holds [1,4] It is well known that the family of operators {A(t)} 0≤t≤T generate a two parameters operators {U(t, s)} 0≤s≤t≤T , see e.g.[9] or [1,Page 832].The evolution equation ( 1) can be written as follows The following theorem provides the well posedness of problem (1) (or ( 8)). Theorem Moreover, if Assumption 2.1 is fulfilled, then the following space regularity holds1 (−A(s))

Finite element discretization
Let T h be a triangulation of Λ with maximal length h.Let V h ⊂ V denotes the space of continuous and piecewise linear functions over the triangulation T h .We defined the projection For any t ∈ [0, T ], the discrete operator A h (t) : The space semi-discrete version of problem (8) consists of finding u h (t) ∈ V h such that For t ∈ [0, T ], we introduce the Ritz projection R h (t) : It is well known (see e.g.[6, (3.2)] or [1]) that the following error estimate holds The following error estimate also holds (see e.g.[6, (3.3)] or [1]) for any r ∈ [1, 2] and v ∈ V ∩ H r (Λ), where is the time derivative of R h .According to the generation theory, A h (t) generates a two parameters evolution operator {U h (t, s)} 0≤s≤t≤T , see e.g.[1,Page 839].Therefore the mild solution of (13) can be written as follows In the rest of this paper, C ≥ 0 stands for a constant indepemdent of h, that may change from one place to another.It is well known (see e.g.[1, Chapter III, (12.3) & (12.4)]) that for any 0 ≤ γ ≤ α ≤ 1 and 0 ≤ s < t ≤ T , the following estimates hold2 3. Main result

Preliminaries result
We consider the following linear homogeneous problem: find w ∈ D ⊂ V such that The corresponding semi-discrete problem in space is: find The following lemma will be useful in our convergence analysis.
Proof.We split the desired error as follows Using the definition of R h (t) and P h (( 11)-( 12)), we can prove exactly as in [4] that One can easily compute the following derivatives Endowing V and the linear subspace V h with the norm .H 1 (Λ) , it follows from (15) that . By the definition of the differential operator, it follows that for all t ∈ [0, T ] and it follows from (24) that Adding and subtracting P h A(t)w(t) in (23) and using (22), it follows that From (23), the mild solution of θ is given by Splitting the integral part of (27) in two and integrating by parts the first one yields Using the expression of θ(τ ), ρ(τ ) (see ( 21)) and the fact that u h (τ ) = P h v, it holds that θ(τ ) + P h ρ(τ ) = 0. Hence (28) reduces to Taking the norm in both sides of (29) and using (18) yields Using ( 15) and ( 16), it holds that Note that the solution of ( 19) can be represented as follows.

Error estimate of the semilinear problem under global Lipschitz condition
Theorem 3.1.Let Assumptions 2.1 and 2.2 be fulfilled.Let u(t) and u h (t) be defined by (9) and (17) respectively.Then the following error estimate holds If in addition the nonlinearity F satisfies the linear growth condition v ∈ H, then the following optimal error estimate holds where β is defined in Assumption 2.1.
Remark 3.1.Note that the hypothesis F (t, v) ≤ C v is not too restrictive.An example of class of nonlinearities for which such hypothesis is fulfilled is a class of func- Concrete examples are operators of the form Remark 3.2.It is possible to obtain an error estimate without irregularities terms of the form t −1+β/2 with a drawback that the convergence rate will not be 2, 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 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 Using Lemma 3.1 with r = 2 and γ = β yields Using Assumption 2.2, (18) and ( 10) yields If 0 ≤ t ≤ h 2 , then using (18) easily yields I 1 ≤ Ch 2 + t 0 u(s) − u h (s) ds.If 0 < h 2 ≤ t, using Lemma 3.1 (with r = 2 and γ = 0), and splitting the second integral in two parts yields Substituting ( 46) and ( 44) in ( 43) and applying Gronwall's lemma proves (41).To prove (55), we only need to re-estimate the term (with r = 2 and γ = δ) and ( 10), following the same lines as above one easily obtain Let us now estimate I 3 under the hypothesis F (t, v) ≤ C v .Using Assumption 2.2, (10) and exploiting the mild solution (17) one easily obtain for some ǫ ∈ (0, 1) and any s, t ∈ [0, T ].Using Lemma 3.1 (with r = 2 and γ = 0), triangle inequality and (47) yields Hence the new estimate of I 1 is given below Substituting ( 48) and ( 44) in (43) and applying Gronwall's lemma proves (55) and the proof of Theorem 3.1 is completed.

Error estimate of the semilinear problem under polynomial growth condition
In this section, we take β ∈ d 2 , 2 .We make the following assumptions on the nonlinearity.
Assumption 3.1.there exist two constants and L 1 , c 1 ∈ [0, ∞) such that the nonlinear function F satisfies the following Let us recall the following Sobolev embedding (continuous embedding).
It is a classical solution that under Assumption 3.1 (8) has a unique mild solution u satisfying3 u ∈ C [0, T ], D (−A(0)) β , see e.g.[9].Hence using the Sobolev embbeding (51), it holds that Theorem 3.2.Let u(t) and u h (t) be solution of ( 8) and (13) respectively.Let Assumptions 2.1 and 3.1 be fulfilled.Then the following error estimate holds If in addition there exists c 1 , c 2 ≥ 0 such that the nonlinearity F satisfies the polynomial growth condition then the following optimal error estimate holds Proof.The proof goes along the same lines as that of Theorem 3.  In fact, let us assume without loss of generality that ϕ is polynomial of degree l > 1, that is Note that the proofs in the cases l = 0, 1 are obvious.For any u ∈ H ∩ C(Λ, R), using traingle inequality and the fact This completes the proof of (49).The proof of (50) is similar to that of (49) by using the following well known fact Remark 3.6.If in Remark 3.5 we take the constant term of ϕ to be 0, then the hypothesis (54) is fulfilled.