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.
on the Hilbert space , where is an open bounded subset of , with smooth boundary. The second order differential operator is given by
where and are smooth coefficients. Also, there exists , such that
Moreover, satisfies the following ellipticity condition
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.
The initial data belongs to , .
The nonlinear function is Lipschitz continuous, i.e. there exists a constant such that
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
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
where is a positive constant, independent of . Note that is bounded in ((Suzuki, , Chapter III, (11.13))), so the following operator is well defined
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
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
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 ).
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
For any , the discrete operator is defined by
The space semi-discrete version of problem (8) consists of finding such that
For , we introduce the Ritz projection defined by
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
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.
3 Main result
3.1 Preliminaries result
We consider the following linear homogeneous problem: find such that
The corresponding semi-discrete problem in space is: find such that
The following lemma will be useful in our convergence analysis.
Proof. We split the desired error as follows
One can easily compute the following derivatives
From (23), the mild solution of is given by
Splitting the integral part of (27) in two and integrating by parts the first one yields
Note that the solution of (19) can be represented as follows.
Therefore it holds that
Taking the derivative with respect to in both sides of (32) yields
Using the estimate
it follows from (39) that
3.2 Error estimate of the semilinear problem under global Lipschitz condition
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
where is defined in Assumption 2.1.
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 .
Using Lemma 3.1 with and 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 . 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
Hence the new estimate of is given below
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.
there exist two constants and such that the nonlinear function 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 satisfying333This remains true if is replaced by its discrete version . , see e.g. Sobolev . Hence using the Sobolev embbeding (51), it holds that
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
Let be polynomial of any order. The nonlinear operator is defined as the Nemytskii operator
is a concrete example satisfying Assumption 3.1.
In fact, let us assume without loss of generality that is polynomial of degree , that is
Note that the proofs in the cases are obvious. For any , using traingle inequality and the fact , , we obtain