Abstract
We consider abstract quasilinear evolution equations of Sobolev type in a Hilbert setting. We propose two fully discrete schemes and prove some error estimates under minimal assumptions. Various examples that enter into our abstract framework are considered, for each of them our theoretical results are confirmed by several numerical experiments.
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1 Introduction
The purpose of our paper is to study different numerical schemes for the abstract quasilinear Sobolev equation
where \(A_1(t,u)\) is an isomorphism from a Hilbert space V into its dual \(V'\), while \(A_2(t,u)\) is a bounded operator from V into \(V'\) (plus some assumptions specified below).
The linear or semilinear case, corresponding to the situation when \(A_i(t,u)\) do not depend neither on t nor on u, will retain some particular interests.
Such problems are interesting not only because they are generalizations of a standard parabolic problem but also because they arise naturally in a large variety of applications (model of fluid flow in fissured porous media [4], two-phase flow in porous media with dynamical capillary pressure [12, 20], heat conduction in two-temperature systems [9, 42] and shear in second order fluids [11, 41]).
Existence results for such problems are proved for semi-linear or non-autonomous equations (i.e., the case when \(A_1\) and \(A_2\) depends only on t) in [7, 16, 27, 38,39,40] for instance, where the authors exploit the fact that \(A_1\) is invertible. This allows to reduce the problem into a first order evolution equation (see (2.3) below) with a bounded (non-autonomous) operator and existence results easily follow. The same idea is here used to show existence results in the quasilinear situation by using the results from [23].
A large numbers of papers are devoted to the discretization of pseudoparabolic equations. Crank-Nicolson/explicit multistep approximation in time is combined with a finite element method in [2, 8, 15, 28, 43], with a Petrov-Galerkin method in [3, 14] and with a discontinuous Galerkin method in [17, 31, 32]. The discretization along characteristics is applied in [34], while a Fourier-Galerkin method is used in [35]. In all these references, the operators \(A_1\) and \(A_2\) are (eventually non linear) second order elliptic operators. Hence in the spirit of [8] our main goal is to perform a general analysis for a fully discrete scheme by combining some error estimates of explicit semi-discrete schemes in time of ordinary differential equations (adapted to Hilbert valued equations) with new error estimates of the corresponding fully discrete schemes based on some ”regularity” assumptions (see assumption \(\mathbf {H_7}\)) and interpolation error estimates. Altogether, if \(U_{n,h}\) is the fully discrete approximation of the solution u at time \(t_n\) obtained by the Euler scheme or the Runge–Kutta scheme of order 2, we prove the error estimate
for all \(n=1,\ldots , N\), where \(p=1\) (resp. \(p=2\)) for the Euler (resp. Runge–Kutta) scheme and q(s) is related to our abstract assumptions (but in practice it depends on the regularity of the initial datum and the chosen finite element space), and C is a positive constant independent of h and \(\Delta t\). Similarly to [8, see p. 14], the operator \(\mathcal {A}(t,u)=A_1(t,u)^{-1}A_2(t,u)\) (involved in (2.3)) satisfies an appropriated Lipschitz property (see Corollary 2.3 below), hence its associated evolution problem is nonstiff. Due to this property, we do not have to impose mesh restrictions, like the CFL one. Note that, contrary to [8], our approach does not require any smallness restriction on the time step and on the meshsize.
We finally illustrate our abstract framework by various examples, like the case when the \(A_i\)’s are (linear, non-autonomous, quasilinear) second order differential operators in smooth and non-smooth domains. In each case, new analytic results are proved to check that our general assumptions hold and some numerical tests that confirm the orders of convergence are presented.
In the whole paper, the norm of V will be denoted by \(\Vert \cdot \Vert \) and we will write \(a \lesssim b\), for the existence of a generic positive constant C that can depend on the final time T and on the norm of the data but is always independent of a, b, of the time step \(\Delta t\) and the meshsize parameter h such that \(a \le C b\).
The paper is organized as follows. In Sect. 2, we give the basic assumptions that allow to obtain existence results. Section 3 is devoted to the introduction of the semi-discrete and the fully discrete problems and to the proof of error estimates. Some illustrative examples and numerical tests are presented in Sects. 4 and 5 for semi-linear and quasi-linear equations.
2 Existence results
We associate to each operator \(A_i(t,u):V\rightarrow V',\)\(t\in [0,T], u\in V, i=1,2,\) a bilinear form \(a_i(t;u;\cdot ,\cdot ),\) via the relation
In this section, we give some (local) existence results for problem (1.1) under the following assumptions.
- \(\mathbf {H_1}\) :
-
(uniform continuity of \(a_i(t;u;\cdot ,\cdot )\) with respect to t and u) for \(i=1,2,\) there exists a constant \(M_i>0\) independent of t and u such that for all \(t\in [0,T]\) and \(u, v,w\in V,\)
$$\begin{aligned} |a_i(t;u;v,w)|\le M_i\Vert v\Vert \Vert w\Vert . \end{aligned}$$ - \(\mathbf {H_2}\) :
-
(uniform coerciveness of \(a_1(t;u;\cdot ,\cdot )\) with respect to t and u) there exists a constant \(\alpha >0\) independent of t and u such that for all \(t\in [0,T]\) and \(u, v\in V,\)
$$\begin{aligned} a_1(t;u;v,v)\ge \alpha \Vert v\Vert ^2. \end{aligned}$$(2.2)
The hypothesis \(\mathbf {H_1}\) is equivalent to the uniform (in t and u) continuity of \(A_i(t,u)\) from V into \(V'\), with \(\Vert A_i(t,u)\Vert _{\mathcal {L}(V,V')}\le M_i\), for any \(t\in [0,T]\) and \(u\in V\); while the hypothesis \(\mathbf {H_2}\) and Lax-Milgram’s lemma guarantees that the operator \(A_1(t,u), t\in [0,T], u\in V\) is an isomorphism from V into \(V'\), with \(\Vert A_1(t,u)^{-1}\Vert _{\mathcal {L}(V',V)}\le \frac{1}{\alpha }\), for any \(t\in [0,T]\) and \(u\in V\).
As the operator \(A_1(t,u), t\in [0,T], u\in V\) is invertible, we can compose the two sides of the first identity of (1.1) by \(A_1(t,u)^{-1}\) and obtain the equivalent problem
where \(\mathcal {A}(t,u)=A_1(t,u)^{-1}A_2(t,u)\) is a bounded operator from V into itself (uniformly with respect to t and u owing to \(\mathbf {H_1}\) and \(\mathbf {H_2}\) with \(\Vert \mathcal {A}(t,u)\Vert _{\mathcal {L}(V,V)}\le \frac{M_2}{\alpha }\), for any \(t\in [0,T]\) and \(u\in V\)) and \(g(t,u)= A_1(t,u)^{-1}f(t,u).\) This problem enters into the framework of Kato’s theory [23, Theorem 6], hence it suffices to check that the assumptions of this theorem are satisfied to obtain a local existence result. This is made under some additional assumptions on the sesquilinear forms \(a_i\) and on f. Before let us make the following definition.
Definition 2.1
Let E and F be two Hilbert spaces. A mapping \(f:[0,T]\times E\longrightarrow F\) is called (E, F)-Lipschitz continuous with respect to the second variable uniformly in t, if there exists a positive constant L independent of t such that
If \(E=F,\) we will say that f is E-Lipschitz continuous with respect to the second variable uniformly in t.
For a fixed open ball W of V, we now introduce the next assumptions:
- \(\mathbf {H_3}\) :
-
there exists \(\gamma \in (0,1]\) such that for \(i=1\) or 2,
$$\begin{aligned} \big |a_i(t;y;u,v)\!-\!a_i(s;z;u,v)\big |\,\lesssim \, \big (|t\!-\!s|^\gamma \!+\!\Vert y\!-\!z\Vert \big )\Vert u\Vert \Vert v\Vert ,\; \forall y, z, u, v\in V,t, s\in [0,T]. \end{aligned}$$(2.4) - \(\mathbf {H_4}\) :
-
f is \((V,V')\)-Lipschitz continuous with respect to the second variable uniformly in t.
- \(\mathbf {H_5}\) :
-
f is bounded from \([0,T]\times V\) into \(V'\):
$$\begin{aligned} \Vert f(t,v)\Vert _{V'}\lesssim 1, \quad \forall \,t\in [0,T],\;\forall \,v\in V. \end{aligned}$$ - \(\mathbf {H_6}\) :
-
For all \(v\in W,\) the mapping \(t\rightarrow f(t,v)\) is continuous from [0, T] into \(V'.\)
We now give some consequences of these assumptions.
Lemma 2.2
Under the hypotheses \(\mathbf {H_1}\)–\(\mathbf {H_3},\) the function \((t,v)\rightarrow A_1(t,v)^{-1}\) is Lipschitz continuous on \([0,T]\times V\) for the norm of \(\mathcal L(V',V)\) uniformly in t, v, namely
Proof
Let \(v,v_0\in V\) and \(t,t_0\in [0,T]\) be arbitrarily fixed. Then by definition we have
Now for \(h\in V', h\ne 0\), if we set \(\phi _1=A_1(t,v)^{-1}h,\) and \(\phi _2=A_1(t_0,v_0)^{-1}h,\) then
which yields
Choosing \(\psi =\phi _1-\phi _2,\) and using the hypotheses \(\mathbf {H_2}\) and \(\mathbf {H_3},\) we obtain
which proves the estimate (2.5). \(\square \)
Corollary 2.3
Under the hypotheses \(\mathbf {H_1}\)–\(\mathbf {H_3},\) the function \((t,v)\rightarrow \mathcal {A}(t,v)\) is Lipschitz continuous on \([0,T]\times V\) for the norm of \(\mathcal L(V)\) uniformly in t, v, namely
As a consequence, we have
Proof
By definition, for \(t, t_0\in [0,T], u, v, v_0\in V\) arbitrarily fixed with \(u\ne 0\), we have
Since the assumption \(\mathbf {H_3}\) for \(i=2\) is equivalent to
we conclude that (2.6) holds owing to (2.5) and the assumptions \(\mathbf {H_1}\) and \(\mathbf {H_2}\). \(\square \)
Corollary 2.4
Under the hypotheses \(\mathbf {H_1}\) to \(\mathbf {H_6}\), the function \(g(t,u)=A_1(t,u)^{-1}f(t,u)\) satisfies the next properties:
-
1.
g is bounded on \([0,T]\times V\), i. e.,
$$\begin{aligned} \Vert g(t,v)\Vert \lesssim \; 1,\;\;\forall u,v\in V,\;\forall t\in [0,T]. \end{aligned}$$ -
2.
g is V-Lipschitz continuous with respect to the second variable uniformly in t,
$$\begin{aligned} \Vert g(t,v)-g(t,v_0)\Vert \lesssim \;\Vert v-v_0\Vert ,\;\;\forall v, v_0\in V,\;\forall t\in [0,T]. \end{aligned}$$(2.8) -
3.
for all \(v\in W\), the mapping \(t\rightarrow g(t,v)\) is continuous from [0, T] into V.
Proof
1. Direct consequence of the hypotheses \(\mathbf {H_2}\) and \(\mathbf {H_5}.\)
2. Let \(v, v_0\in V\) and \(t, t_0\in [0,T]\), then by the assumption \(\mathbf {H_2}\), we may write
Hence by our assumption \(\mathbf {H_5}\) and (2.5), we obtain
If in particular \(t_0=t\), this estimate and the Lipschitz property of f then yield (2.8).
3. If \(v_0=v\in W\) in the estimate (2.9), we get
and the continuity property on g follows from the assumption \(\mathbf {H_6}\). \(\square \)
We are ready to prove our existence result.
Theorem 2.5
Fix an open ball W of V and suppose that \(u_0\in W.\) Under the assumptions \(\mathbf {H_1}\)–\(\mathbf {H_6}\), there exists \(T'\in (0, T]\) such that problem (2.3) (or (1.1)) admits a unique strong solution u in \([0,T']\), i.e., with the regularity \(u\in C([0,T'];W)\cap C^{1}([0,T'];V).\)
Proof
We apply Theorem 6 from [23] with the Hilbert space \(X=Y=V\) and the fixed open ball W from the statement. The assumptions (A1), (A2) and (A4) of this Theorem are trivially satisfied because the operators \(\mathcal {A}(t,v)\) are bounded in V, the assumption (A3) holds owing to Corollary 2.3, while assumption (f1) holds owing to Corollary 2.4. \(\square \)
Remark 2.6
Note that in the linear or semilinear case and under the assumption\(f\in C^{1}([0,T]\times V;V')\) and \(\mathbf {H_1}\)–\(\mathbf {H_2}\), Theorem 6.1.5 of [33] guarantees the existence of a global solution \(u\in C^{1}([0,T];V)\) for any initial data in V.
We end up this section with the following comment. If we suppose that there exists another Hilbert space H such that V is continuously embedded into H (denoted by \(V\hookrightarrow H\)) and such that V is a dense subspace of H, then we can introduce the restriction of \(A_i(t,u)\) to H (that, for shortness, is still denoted by \(A_i(t,u)\)), namely we can define the unbounded operator from H into itself by
and
3 Discretizations of the problem
3.1 Explicit semi-discretization in time
We notice that problem (1.1) can be equivalently written as
where \(F(t,u)=g(t,u)-\mathcal {A}(t,u)u\), which is Hilbert-valued nonlinear ordinary differential equation. Since in our case, \(F(\cdot ,\cdot )\) is bounded, we can use standard explicit schemes, like the Euler or Runge–Kutta methods as in the case of finite-dimensional ODE. More precisely, we now consider a regular subdivision \((t_i=i\Delta t)_{i=0}^{N}\) of the interval \([0,T'],\) where \(T'\) is the life time of u, \(N\in \mathbb {N}^{*}\) and \(\Delta t=\frac{T'}{N}\) the time step. Given a continuous function \(\phi :[0,T]\times V\times [0,\Delta t]\longrightarrow V\), starting from \(u(t_0=0)=u_0\), we try to estimate the solution u of (3.1) at the points \((t_{n+1}), n=0,\ldots ,N-1\), by estimating step by step the values of \(u(t_{n+1})\) using the variation of constants formula
Here we restrict ourselves to a one step method that consists to approach the expression \(\int _{t_n}^{t_{n+1}}F(\tau ,u(\tau ))d\tau \) by \(\Delta t\phi (t_n,u(t_n),\Delta t),\) i.e., the approximated solution of problem (3.1) is given by
The convergence of this numerical scheme is based on the estimation of the local consistency error.
Definition 3.1
The local consistancy error \(E_l\) relative to the exact solution u of (3.1) is defined by
The next Theorem is a direct generalization of a well-known result for ODE in the form \(u_t=F(t,u),\) where F has values in \(\mathbb {R}^k, k\in \mathbb {N}^*\) (see for example Theorem 3.5 of [19]).
Theorem 3.2
Let the assumptions of Theorem 2.5 be satisfied (or Remark 2.6 in the linear or semilinear case). Let u be the exact solution of problem (3.1). Suppose that \(\phi \) is locally Lipschitz continuous with respect to the second variable uniformly in t, i.e., there exists a positive constant L independent of t and \(\Delta t\) such that
and suppose that there exists \(p\in \mathbb {N}\) such that the local errors satisfy
Then the global errors \(e_n=u(t_n)-U_n\) satisfy
Proof
We have from (3.3),
Then, from (3.2), the Lipschitz property on \(\phi \) and the fact that \(u\in C([0,T'],W),\) we obtain
By induction on n, we deduce that
Since \(e_0=0\), we find that (3.5) holds. \(\square \)
Lemma 3.3
Assume that \(u_0\in W\). Under the assumptions \(\mathbf {H_1}\)–\(\mathbf {H_5}\), the function F is locally Lipschitz with respect to the second variable uniformly in t, i.e., for all \(t\in [0,T],\)
Proof
By definition of F, it is easy to see that
The result follows thanks to Corollary 2.2, (2.7) and the fact that \(u\in W\). \(\square \)
Remark 3.4
In the linear or semilinear case, the assumptions \(\mathbf {H_1} \), \(\mathbf {H_2} \) and \(\mathbf {H_4} \) guarantee that F is V-Lipschitz continuous with respect to the second variable uniformly in t.
We now concentrate on two particular schemes.
3.1.1 Explicit Euler scheme
This scheme corresponds to the choice \(\phi (t,u,\Delta t)= F(t,u), \) and then takes the form
In this case, we have the next error estimate.
Proposition 3.5
Under the assumptions of Theorem 2.5, assume that the solution u of (1.1) has the extra regularity \(C^{2}([0,T'];V)\). If \(U_n\) is the approximated solution given by the explicit Euler scheme (3.7), then the local errors \(E_l\) satisfy the following estimate
Furthermore the global errors satisfy
Proof
By a Taylor development with integral remainder at order 1, we have
Consequently, one has
and we conclude that (3.8) holds by our assumption.
For the second assertion, we simply notice that Lemma 3.3 guarantees that\(\phi (t,u,\Delta t)=F(t,u)\) satisfies (3.4) and we conclude by Theorem 3.2. \(\square \)
Remark 3.6
In the linear or semilinear case, it is not difficult to check that the assumption\(f\in C^{1}([0,T']\times V;V^{'})\) implies the extra regularity \(u\in C^{2}([0,T'];V)\). In the general situation, we further need that the mapping \((t,v)\rightarrow A_i(t,v), i=1,2,\) is Fréchet differentiable on \([0,T']\times V,\) with
3.1.2 Heun’s scheme (or Runge–Kutta of order 2)
This scheme corresponds to the choice
and then may be written as
Proposition 3.7
Under the assumptions of Theorem 2.5, assume that the solution u of (1.1) has the extra regularity \(C^3([0,T'];V)\). If \(U_n\) is the approximated solution given by the Runge–Kutta scheme, then
As a consequence, we have
Proof
The first assumption follows by a Taylor development with integral remainder at the order 2. Thanks to Lemma 3.3, we easily check that
which implies that \(\phi \) satisfies (3.4). The second assertion then follows from Theorem 3.2. \(\square \)
Remark 3.8
In the linear or semilinear case, the assumption \(f\in C^2([0,T']\times V;V^{'})\) implies the extra regularity \(u\in C^3([0,T'];V)\). In the general situation, we further need that the mapping \((t,v)\rightarrow A_i(t,v), i=1,2,\) is twicely Fréchet differentiable on \([0,T']\times V,\) with
3.2 Fully discrete scheme
For a positive parameter h (that plays the rule of a mesh size), we suppose given a finite dimensional subspace \(V_h\) of V and build a fully discrete approximation of problem (1.1). For that purpose, let us introduce some useful notations. For an arbitrary element \(u_h\) in \(V_h\), we consider the approximation \(A_{i,h}(t,u_h)\) of \(A_i(t,u_h),i=1,2\), defined by
For further uses, for any \(t\in [0,T]\) and any \(u\in V,\) we define the orthogonal projection \(P_h(t,u)\) associated with the bilinear form \(a_1(t;u;\cdot ,\cdot ),\) i.e., for any \(v\in V,P_h(t,u)v\in V_h\) is the unique solution of
Similarly we introduce the orthogonal projection \(Q_h\) in V on \(V_h\) associated with the inner product \((\cdot ,\cdot )_V.\)
We first consider the discrete (in space) version of (1.1), namely we look for \(u_h\in C^1([0,T], V_h)\) solution of
where \(f_{h}(t,u_h)=I_{h}f(t,u_h)\) and \(I_{h}: V' \rightarrow V_{h}'\) is the linear and continuous operator defined by
As in the continuous case, the operator \(A_{1,h}(t,u_h), t\in [0,T], u_h\in V_h\) being invertible, this problem is then equivalent to
where \(\mathcal {A}_h(t,u_h)=A_{1,h}(t,u_h)^{-1}A_{2,h}(t,u_h)\) and \(g_h(t,u_h)=A_{1,h}(t,u_h)^{-1}f_h(t,u_h).\) The next Lemma shows that the operator \(\mathcal {A}_h(t,u_h)\) is bounded (uniformly with respect to h) from \(V_h\) into itself.
Lemma 3.9
Under the hypotheses \(\mathbf {H_1}\) and \(\mathbf {H_2},\) for any \(t\in [0,T]\) and \(v_h\in V_h\), one has
Proof
Let us fix \(t\in [0,T]\) and \(v_h\in V_h.\) Then by definition we have
For a fixed \(w_h\in V_h,w_h\ne 0\), we set \(v'_h=A_{1,h}(t,v_h)^{-1}A_{2,h}(t,v_h)w_h,\) then owing to (3.11), \(v_h'\in V_h\) is the unique solution of
Taking \(\varphi _h=v'_h\) and using the hypotheses \(\mathbf {H_1}\) and \(\mathbf {H_2},\) we find that
which implies (3.14). \(\square \)
Note that problem (3.13) can be equivalently written as an ODE in \(V_h\):
where \(F_h(t,u_h)=g_h(t,u_h)-\mathcal {A}_h(t,u_h)u_h.\) Therefore its approximation by an explicit Euler scheme or by a Runge–Kutta scheme of order 2, takes the respective forms
or
Remark 3.10
For any \(u_h\in V_h,\) the hypotheses \(\mathbf {H_1}\) and \(\mathbf {H_2}\) yield
Indeed as \(A_{1,h}(t,u_h)g_{h}(t,u_h)=I_h f(t,u_h),\) we obtain
On one hand, the definition of \(P_h(t,u_h)\) implies that
on the other hand, by the definition of the bilinear form \(a_1(t;u_h;\cdot ,\cdot ),\) we find that
This proves (3.18) by Lax–Milgram’s lemma.
Our error analysis of the fully discrete scheme is based on the following assumptions.
- \(\mathbf {H_7}\) :
-
There exist two Hilbert spaces \(D_s\) and \(\tilde{D}_{s-1}\) (\( s\ge 1\) being a parameter that could take different occurrence or not) such that \( D_s\hookrightarrow V\) and such that \(A_1(t,u)\) is an isomorphism from \(D_s\) into \(\tilde{D}_{s-1}\), while \(A_2(t,u)\) is only supposed to be bounded from \(D_s\) into \(\tilde{D}_{s-1},\) with
$$\begin{aligned} \Vert A_1(t,u)^{-1}\Vert _{\mathcal L(\tilde{D}_{s-1},D_s)}+ \Vert A_2(t,u)\Vert _{\mathcal L(D_s,\tilde{D}_{s-1})} \lesssim 1, \end{aligned}$$for all \(t\in [0,T]\) and \(u\in V\).
- \(\mathbf {H_8}\) :
-
For each parameter s from assumption \(\mathbf {H_7}\), there exists a positive real number q(s) such that
$$\begin{aligned} \Vert \varphi -Q_h\varphi \Vert \lesssim h^{q(s)}\Vert \varphi \Vert _{D_s}, \;\;\;\forall \varphi \in D_s. \end{aligned}$$(3.19)
In practice, under the assumption that \(V\hookrightarrow H\), \(D_s\) corresponds to the domain of powers of \(A_1(t,u)\) or a subdomain of it (hence \(\tilde{D}_{s-1}=A_1(t,u)D_s\)), while the estimate (3.19) follows from an interpolation error estimate. In particular for \(s=1\), we can chose \(D_1=D(A_1(t,u))\) and \(\tilde{D}_{0}=H\), if \(D(A_1(t,u))\) is independent of t and u, and if we can check the above assumptions. We refer to Sects. 4 and 5 for some concrete illustrations.
Now, if we denote by
the norm on V associated with the bilinear form \(a_1(t;v;\cdot ,\cdot ),\) by the continuity and the uniform coercivity of \(a_1(t;v;\cdot ,\cdot )\) (hypotheses \(\mathbf {H_1}\) and \(\mathbf {H_2}\)), we notice that
Proposition 3.11
Let the hypotheses \(\mathbf {H_1}\)–\(\mathbf {H_8}\) be satisfied. If we suppose that \(f\in C([0,T]\times V;\tilde{D}_{s-1})\), then for all \(t\in [0,T]\) and \((v,v_h)\in D_s\times V_h,\) we have
Proof
Fix \(t\in [0,T]\) and \((v,v_h)\in D_s\times V_h\). By the triangular inequality, we have
We start by the estimating the first term of this right-hand side. The identity (3.18) yields
On one hand using (3.20), we obtain
where the last estimate is due to Corollary 2.4, 2. On the other hand, using again (3.20) we obtain
Owing to the hypothesis (3.19), we find
For the second term, we may write
Corollary 2.3 directly furnishes
Further owing to (3.20), we have
As \(\mathcal {A}(t,v_h)v\in D_s\) due to the assumption \(v\in D_s,\) we deduce from the estimate (3.19) that
It then remains to estimate the last term of the right-hand side of (3.22). For that purpose, setting \(w=\mathcal {A}(t,v_h)v\) and \(w_h=\mathcal {A}_h(t,v_h)v_h,\) by the definition of \(a_1\) and \(a_2,\) we have
and by the definition of \(P_h(t,v_h),\) we also have
Hence
Choosing in (3.23) \(\psi _h=P_h(t,v_h)w-w_h,\) we obtain owing to the hypotheses \(\mathbf {H_1}\) and \(\mathbf {H_2}\)
Consequently
Altogether we have shown that (3.21) is valid. \(\square \)
Lemma 3.12
Suppose that \(u_0\in D_s,\) that \(f\in C([0,T]\times V;\tilde{D}_{s-1})\) is \((V,\tilde{D}_{s-1})\)-Lipschitz continuous with respect to the second variable uniformly in t, and that \(\mathbf {H_7}\) hold. Then the sequence \((U_n)_{n=1}^{N}\) defined by (3.10) is (uniformly) bounded in \(D_s,\) i.e.,
Proof
By construction (see (3.10)), we have
We then estimate each term of this right-hand side separately by using the assumptions that \(A_1(t,v)^{-1}\) is (uniformly) bounded from \(\tilde{D}_{s-1}\) into \(D_s,\) that \(\mathcal A(t,v)\) is (uniformly) bounded in \(D_s\) and taking into account the Lipschitz hypothesis on f.
For the first term, by the triangle inequality, for any \(u\in D_s\) we have
Since \(F(t_n,0)=A_1(t_n,0)^{-1}f(t_n,0)\), we directly get
Furthermore for any \(u\in D_s,\) and for any \(t\in [0,T]\), we can write
By the Lipschitz property on f, we then conclude that
These two estimates in (3.25) leads to
To estimate the second term of the right-hand side of (3.24), writting
and using (3.26) and (3.27), we obtain
Therefore (3.28) allows to conclude that
Using this estimate and (3.28) in (3.24) leads to
By iteration, we find
As for all \(0\le k\le N\)
the result follows since f is continuous from \([0,T]\times V\) into \(\tilde{D}_{s-1}.\)\(\square \)
Lemma 3.13
Let the hypotheses \(\mathbf {H_1}\) to \(\mathbf {H_8}\) and the hypotheses of Lemma 3.12 hold. Let \(U_n\) (resp. \(U_{n,h}\)) be the approximated solution given by (3.10) (resp. (3.17)). Then the error \(e_{n,h}=U_n-U_{n,h}\) is bounded as follows
Proof
By the triangular inequality, we can write
As \(U_n\in D_s,\) for all \(1\le n\le N,\) Proposition 3.11 and Lemma 3.12 lead to
Using again this proposition and the previous estimate, we find
Consequently
By iteration we deduce that
As \(u_0\in D_s,\) our hypothesis (3.19) guarantees that
By the Lipschitz continuity of f and since \(D_s\) is continously embedded into V, we obtain
Similarly one has
With the help of the estimate (3.28), we obtain
Inserting this estimate, as well as (3.31) and (3.32) into (3.30), we deduce owing to Lemma 3.12 and the continuity of f from \([0,T]\times V\) into \(\tilde{D}_{s-1}\) that (3.29) is valid. \(\square \)
Remark 3.14
Under the same hypotheses of Lemma 3.13, if the Euler scheme (3.16) is used to approximate the solution of problem (1.1), then the error estimate (3.29) remains valid.
Corollary 3.15
Under the hypotheses \(\mathbf {H_1}\) to \(\mathbf {H_8}\), we suppose that \(u_0\in W\cap D_s,\)\(f\in C([0,T]\times V; \tilde{D}_{s-1}),\) that f is \((V,\tilde{D}_{s-1})\)-Lipschitz continuous with respect to the second variable uniformly in t. Assume that the solution u of problem (1.1) exists and belongs to \(C^{p+1}([0,T'];V)\), \(p=1\) or 2. Let \(U_{n,h}\) be its approximated solution given by (3.16) for \(p=1\) (resp. (3.17) for \(p=2\)). Then we have the global error estimate
Proof
Direct consequence of Proposition 3.5 and Remark 3.14 for the Euler scheme, and of Proposition 3.7 and Lemma 3.13 for the Runge–Kutta scheme. \(\square \)
4 Applications to particular semi-linear equations
4.1 Elliptic operators of order two: the regular case
Let \(\Omega \) be a bounded domain of \(\mathbb {R}^{d}, d\ge 1\) with a Lipschitz boundary. For \(i=1,2,\) let \(L_i\) be two elliptic operators of order two of the form
where \(a^{(i)}_{k,\ell }=a^{(i)}_{\ell , k}\in C^{0,1}(\bar{\Omega })\), \(b^{(i)}_{k}, c^{(i)}\in L^\infty (\Omega ).\) Moreover we suppose that \(L_1\) is strongly elliptic, namely that there exists \(\alpha _*>0\) such that
In this case, we may introduce the continuous bilinear forms \(a_i\) on \(H^{1}_0(\Omega )\times H^{1}_0(\Omega )\) by
Hence their associated operator \(A_i\)
are continuous from \(H^1_0(\Omega )\) into its dual \(H^{-1}(\Omega )\). These operators satisfy the hypotheses from Sect. 2 with \(V=H^1_0(\Omega )\) and \(H=L^2(\Omega )\), if we assume that \(a_1\) is coercive on \(H^1_0(\Omega )\) (that is the case if div \(\mathbf {b}^{(1)}=\sum _{k=1}^d \partial _k b^{(1)}_{k}=0\) and \(c^{(1)}\ge 0\) for example).
Consequently the problem
is well-posedness for an initial datum \(u_0\in H^1_0(\Omega )\) and f continuously differentiable from \([0,T]\times H^1_0(\Omega )\) into \(H^{-1}(\Omega ).\) This system is a semi-linear Sobolev equation in \(\Omega \) that has been analyzed in [2, 15, 28, 31, 32, 39] in some particular situations with Neumann type boundary conditions. Such boundary conditions also enter into our framework by simply replacing \(H^1_0(\Omega )\) by \(H^1(\Omega )\) (and assuming that \(a_1\) is coercive in \(H^1(\Omega )\)).
In order to check the assumptions \(\mathbf {H_7}\) and \(\mathbf {H_8}\), we will characterize the domains of \(A_1\) (as an unbounded operator in H) and of \(A_1^{\frac{3}{2}}\) in some particular situations. In the first case we make use of Kadlec’s result.
Lemma 4.1
Under the previous hypotheses on the coefficients of \(L_i, i=1,2,\) and if \(\Omega \) is convex or has a boundary of class \(C^{1,1},\) then
Proof
For \(i=1,\) or 2, we recall that
Hence \(u\in H^1_0(\Omega )\) belongs to \(D(A_1)\) if and only if there exists \(f\in L^2(\Omega )\) such that
This is equivalent to
where \( h=f-\sum _{k=1}^d b^{(1)}_{k} \partial _k u -c^{(1)} u \) belongs to \(L^2(\Omega )\). Hence owing to Kadlec’s result [22] (see also [18, Thm 3.2.1.3]), we conclude that \(u\in H^2(\Omega )\). This proves the embedding
The inverse embedding being trivial, we have shown that
Clearly we have \(D(A_1)=H^2(\Omega )\cap H^1_0(\Omega )\subset D(A_2),\) and therefore for \(u\in D(A_1),\)\(L_2 u\) belongs to \(L^2(\Omega )\) with
which proves the continuous embedding of \(D(A_1)\) into \(D(A_2)\). \(\square \)
With the help of this result, as a first guess we can take \(D_1=D(A_1)\) and \(\tilde{D}_{0}=H\), since \(A_1\) is an isomorphism from \(D(A_1)\) into H and \(A_2\) is bounded from \(D(A_1)\) into H. The characterization of the domain of \(D(A_1^{\frac{3}{2}})\) and additional assumptions on the coefficients of \(A_2\) allow to build a second choice of pairs \(D_s, \tilde{D}_{s-1}\).
Lemma 4.2
Suppose that the boundary of \(\Omega \) is of class \(C^{2,1},\) that \(a^{(i)}_{k,\ell }=a^{(i)}_{\ell , k}\in C^{1,1}(\bar{\Omega })\), and that \(b^{(i)}_{k}, c^{(i)}\in C^{0,1}(\Omega )\). Then
If furthermore we have
where n is the unit outward normal vector along the boundary and for \(d^{(i)}=(d^{(i)}_l)_{l=1}^d\) is the vector given by
Then \(A_2\) is continuous from \(D(A_1^{3/2})\) into \(H_0^{1}(\Omega )\).
Proof
Let \(u\in D(A_1^{3/2})\). Then there exists \(h\in H^{1}_0(\Omega )\) such that
By [18, Theorem 2.5.1.1, p. 128], we deduce that \(u\in H^{3}(\Omega )\), hence the embedding
The converse embedding being trivial, the first assertion is proved.
Let us go on with the second assertion. Let us then fix \(u\in D(A_1^{3/2})\), then by the regularity of the coefficients of \(L_2\), we directly see that \(A_2 u=L_2u\) belongs to \(H^1(\Omega )\). Hence it remains to show that it is zero on the boundary. For that purpose, we notice that
Hence on the boundary, splitting the gradient of u into its tangential part and its normal one, and recalling that \(u=0\) on the boundary, we have
Our assumption (4.5) then implies that
which finishes the proof since \(L_1u\) is zero on the boundary. \(\square \)
Let us notice that a similar result may remain valid for less regular boundaries. Indeed it holds for instance for a square and for \(L_1\) reduces to the Laplace operator.
Lemma 4.3
If \(\Omega \) is the unit square \((0,1)^2\) of the plane and \(L_1=\Delta ,\) then (4.4) remains valid.
Proof
We use an argument from the proof of Lemma 2.4 in [21]. Let \(v\in D(A_1^{3/2})\hookrightarrow D(A_1)=H^2(\Omega )\cap H_0^{1}(\Omega ),\) then \(g=\Delta v \in H^1_0(\Omega ).\) Hence from elliptic regularity, \(v\in H^3(\Omega \setminus V),\) where V is any neighborhood of the corners. It then remains to show the \(H^3\) regularity near the corners. By symmetry, it suffices to show such a regularity near 0. Let us then fix a radial cut-off function \(\eta \) such that \(\eta =1\) near 0 with \(\text {supp}\;\eta \subset B(0,\frac{1}{2}).\) Consequently \(u=\eta v\) (extended by zero outside its support) belongs to \(H^2((0,\infty )^2)\cap H^1_0((0,\infty )^2)\) and satisfies
We now set
and easily check that
in the distributional sense. But as \(F\in H^1(\mathbb {R}^2)\) by [37, p.85], we conclude that \(U\in H^3(\mathbb {R}^2),\) and consequently \(u\in H^3((0,\infty )^2)\) and finally \(v\in H^3(\Omega ).\)\(\square \)
In conclusion if (4.4) and (4.5) are valid, we can take \(D_{3/2}=D(A_1^{\frac{3}{2}})\) and \(\tilde{D}_{1/2}=D(A_1^{\frac{1}{2}})=H_0^1(\Omega )\).
Now to build a fully discrete scheme, we shall use a finite element method based on a triangulation of \(\Omega \). To this end, we consider a family of meshes \(\left\{ \mathcal T_h\right\} _h\) of \(\Omega \), where each mesh is made of tetrahedral (or triangular) elements K. To simplify the analysis, we assume that the boundary of \(\Omega \) is exactly triangulated, and therefore, we consider curved Lagrange finite elements as described in [5]. Also, for each element K, we denote by \(\mathcal F_K\) the mapping taking the reference element \(\hat{K}\) to K.
With the help of this triangulation \(\mathcal {T}_{h}\), we define the approximation space \(V_h\subset H_{0}^{1}(\Omega )\) by
where \(\mathbb {P}_p(\hat{K})\) stands for the set of polynomials of total degree less than or equal to p.
In this setting, owing to Corollary 5.2 of [5] (see also Theorem 3.2.2 of [10]), the assumption (3.19) is satisfied for \(s=1\) (under the assumption (4.3)) or \(s=\frac{3}{2}\) (under the assumptions (4.4) and (4.5)), with \(q(s)=2s-1\) and the choice \(p\ge q(s)\), i.e., for all \(f\in D_s\), one has
Finally the fully discrete schemes of problem (4.2) can be formulated as follows: The explicit Euler scheme consists in looking for \(U_{n+1,h}\in V_h\) solution of
that allows to compute \(U_{n+1,h}\) by the knowledge of \(U_{n,h}\) and of \(f(t_n,U_{n,h}).\)
Similarly, by the Runge–Kutta method, we look for \(U_{n+1,h}^{*}\in V_h\) solution of
and then \(U_{n+1,h}\in V_h\) solution of
4.2 Elliptic operators of order two: the singular case
We now extend the previous results to the case where the domain \(\Omega \) is a non-smooth two-dimensional domain and the principal part of \(L_1\) and \(L_2\) are piecewise constant. In that case, Lemma 4.1 is no more valid in general (see [24,25,26, 30] for instance), but the use of weighted Sobolev spaces of Kondratiev’s type [18, 36] will allow to put (4.2) into our abstract framework.
Let us start with the definition of the weighted Sobolev spaces in a polygonal domain D of \(\mathbb {R}^2\) (see [36] or [18, Def. 8.4.1.1]).
Definition 4.4
Let r(x) be the distance from a point x of D to the vertices of D. For \(\alpha \in \mathbb {R}\) and \(k\in \mathbb {N}^*\), we define
These spaces are Hilbert spaces equipped with their natural norms:
For any edge e of D, the trace space of \(V^1_\alpha (D)\) onto e is denoted by \(V^{\frac{1}{2}}_\alpha (e)\) (see [29, Thm 1.31]). Note that \(V^{\frac{1}{2}}_\alpha (e)\) has its own definition, see [29, Def. 1.9], in particular we have
We now suppose that \(\Omega \) is a polygonal domain of \(\mathbb {R}^2\) that is partitioned into sub-domains \(\Omega _j\), \(j=1,\ldots , J\), with a positive integer J so that the \(\Omega _j\)’s are disjoint open polygonal domains and that
Let us further denote by \(e_\ell \), \(\ell =1,\ldots , L\), the set of interior edges, namely the set of straight segments that are the intersection of \(\bar{\Omega }_j\cap \bar{\Omega }_{j'}\) with \(j\ne j'\) (hence they are not included into the boundary of \(\Omega \)). Similarly the set \(\mathcal{S}\) of vertices of \(\Omega \) is simply the set of vertices of all \(\Omega _j\)’s.
In the following we need piecewise weighted Sobolev spaces \(\mathcal {V}^k_\alpha (\Omega )\), more precisely, we set
where \(v_j:=v_{|\Omega _j}\) denotes the restriction of v to \(\Omega _j\). Again these spaces are Hilbert spaces equipped with their natural norms.
Now we suppose that the operators \(L_i\) are elliptic of order 2 in the previous form (4.1) but with coefficients \(a^{(i)}_{k,\ell }=a^{(i)}_{\ell , k}\) piecewise regular, in other words the restriction of \(a^{(i)}_{k,\ell }\) to \(\Omega _j\) are regular (\(C^\infty (\bar{\Omega }_j)\)). As before we assume that the bilinear form \(a_1\) associated with \(A_1\) is coercive so that \(A_1\) is an isomorphism from \(H^1_0(\Omega )\) into \(H^{-1}(\Omega )\).
To facilitate the presentation, for \(i=1\) or 2, let us introduce the symmetric matrix \(M_i=(a^{(i)}_{k,\ell })_{k,\ell =1,2}\) and the gradient jumps of u through an edge \(e_\ell \) as follows
when \(e_\ell =\bar{\Omega }_j\cap \bar{\Omega }_{j'}\) and \(n_\ell \) is the unit normal vector along \(e_\ell \) orientated from \(\Omega _j\) to \(\Omega _{j'}\).
We now recall Corollary 4.4 of [30] that is valid in dimension 2 under the assumption \(\gamma <1\) since there exists \(r>1\) such that
Theorem 4.5
Suppose that \(\gamma \in (0,1)\) and the segment \((0,1-\gamma ]\) does not contain singular exponent of \(A_1\) at all corners of \(\Omega \). Then for all \(f\in L^2_\gamma (\Omega )\) and \(h_\ell \in V^{\frac{1}{2}}_\gamma (e_\ell )\), \(\ell =1, \ldots , L\), there exists a unique solution \(u\in H^1_0(\Omega )\cap \mathcal {V}^2_\gamma (\Omega )\) to problem
in the sense that
where the linear form F is given by
Note that F is well-defined on \(H^1_0(\Omega )\) since for all \(\gamma \in (0,1)\), there exists \(r\in (1,2]\) such that
Indeed by (4.9), any \(w\in V^{\frac{1}{2}}_\gamma (e_\ell )\) satisfies
If \(\gamma \le \frac{1}{2}\), we directly obtain \(w\in L^2(e_\ell )\), on the contrary if \(\gamma \in (0,\frac{1}{2})\), as
for all \(t<(\gamma -\frac{1}{2})^{-1}\), owing to Hölder’s inequality we show that \(w\in L^r(e_\ell )\) for some \(r>1\).
This Theorem allows to check the assumption \(\mathbf {H_7}\) with \(D_1=\mathcal {V}_{\gamma }^2(\Omega )\cap H^{1}_0(\Omega )\) and \(\tilde{D}_{0}=L_{\gamma }^2(\Omega )\times \prod _{\ell =1}^L V^{\frac{1}{2}}_{\gamma }(e_\ell )\), for all \(\gamma \in (1-\lambda _a,1)\), when \(\lambda _a\) is the smallest positive singular exponent associated with \(A_1\). Indeed, the previous result asserts that \(A_1\) is an isomorphism from \(D_1\) into \(\tilde{D}_{0}\), therefore it remains to check the boundedness property of \(A_2\):
Lemma 4.6
\(A_2\) is bounded from \(D_1\) into \(\tilde{D}_{0}\).
Proof
Fix \(v\in \mathcal {V}_{\gamma }^2(\Omega )\cap H^{1}_0(\Omega )\), then \(A_2v\) belongs to \(H^{-1}(\Omega )\) and is given by
for all \(w\in H^{1}_0(\Omega )\). As \(b^{(2)}_{k} \partial _k v\) and \(c^{(2)} v\) are in \(L^2(\Omega )\), it remains to transform the first term of this right-hand side. For that purpose, we first fix \(w\in \mathcal{D}(\Omega )\). By an application of Hölder’s inequality, there exists \(r>1\) such that
Therefore Green’s formula on each \(\Omega _j\) yields
As \(\partial _k(a^{(2)}_{k,\ell }\;\partial _\ell ) v\) (resp. 
) belongs to \(L^2_\gamma (\Omega )\) (resp. \(V^{\frac{1}{2}}_\gamma (e_\ell )\)) and since \(H^{1}_0(\Omega )\) (resp. \(H^{\frac{1}{2}}(e_\ell )\)) is embedded into \(L^s(\Omega )\) (resp. \(L^s(e_\ell )\)) for all \(s>1\) and recalling (4.10), (4.12), the previous identity remains valid for all \(w\in H^{1}_0(\Omega )\), owing to Hölder’s inequality. We then deduce that
This ends the proof in view of the regularity of \(L_2 v_j\) and of 
. \(\square \)
In conclusion problem (4.2) is well-posed if we take an initial datum \(u_0\) in \(D_1\) and if f(t, u) is continuous with value in \(\tilde{D}_{0}\). Nevertheless it is well known that the reduction of regularity diminishes the rate of convergence for a standard FEM based on quasi-uniform meshes, but the use of refined meshes near the singular points allows to restore the optimal order of convergence, namely the estimate (3.19) is valid with \(q(1)=1\) (using [36] or [18, Thm 8.4.1.6] on each subdomain \(\Omega _j\)), \(V_h\) being defined by (4.6) with a triangulation that is conform with the partition of \(\Omega \) (i.e., each triangle T of \(\mathcal T_h\) has to be included into one \(\Omega _j\)).
Remark 4.7
Near an exterior vertex, where \(\Omega \) is convex or if the coefficients \(a^{(1)}_{k,\ell }\) are continuous at an interior vertex, then the shift Theorem is valid in standard Sobolev spaces and therefore it is not necessary to take initial data in weighted Sobolev spaces near such vertices but it suffices to take them in \(H^2\). Consequently near such vertices, quasi-uniform meshes can be used.
Remark 4.8
If \(L_2\) is strongly elliptic, then this operator may have one singularity \(S_\mu \) near a vertex, with \(0<\mu <1\), in other words, \(S_\mu \in H^1_0(\Omega )\) behaves like \(r^\mu \) near this vertex, is piecewise regular elsewhere and satisfies
Then the function \(u(x,t)=S_\mu (x)\) is clearly a solution of (4.2) with \(f=g\) and initial datum \(S_\mu \):
This solution is indeed furnished by our abstract framework if we fix the parameter \(\gamma \) appropriately, namely if \( \gamma >\max \{1-\mu , 1-\lambda _a\}. \)
To avoid to take initial data in weighted Sobolev spaces, we will extend our previous framework in the following way. For each interior vertex \(s\in \mathcal{S}_\mathrm{int}\), we fix a smooth cut-off function \(\eta _s\) equal to 1 near s and equal to zero near the other vertices. We then introduce
in other words \(v\in \hat{D}_1\) if and only if there exists \(v_w\in D_1\) and coefficients \(c_s\in \mathbb {R}\), \(s\in \mathcal{S}_\mathrm{int}\) such that
This is a Hilbert space with the inner product
when \(v=v_w+\sum _{s\in \mathcal{S}_\mathrm{int}} c_s\eta _s\) and \(v'=v'_w+\sum _{s\in \mathcal{S}_\mathrm{int}} c'_s\eta _s.\)
The key point is the next result.
Lemma 4.9
The operator \(A_1^{-1} A_2\) is bounded from \(\hat{D}_1\) into itself.
Proof
Take an arbitrary element \(v\in \hat{D}_1\), then it admits the splitting (4.13) and hence
As we have seen in Lemma 4.6 that \(A_2v_w\) is in \(\tilde{D}_{0}\), it remains to show that \(A_2\eta _s\) is in \(\tilde{D}_{0}\) as well. If this is the case, then \(A_2v\) belongs to \(\tilde{D}_{0}\) and we conclude owing to Theorem 4.5.
For \(s\in \mathcal{S}_\mathrm{int}\), let us characterize \(A_2 \eta _s\). By definition we have
for all \(w\in H^{1}_0(\Omega )\). As \(\eta _s\) is regular, we can apply Green’s formula on each \(\Omega _j\) to find
Since \(\eta _s\) is constant near the vertices of \(\Omega \), we deduce that \(L_2 \eta _{s,j}\in L^2_\gamma (\Omega _j)\) and that 
, which shows that \(A_2\eta _s\in \tilde{D}_{0}\). \(\square \)
Corollary 4.10
If \(f\in C([0,T]; L^2(\Omega ))\) and \(u_0\in \{v\in H^1_0(\Omega ): v_j\in H^2(\Omega _j), \forall j=1,\cdots , J\}\), then problem
has a unique solution \(u\in C^1([0,T]; \hat{D}_1)\).
Proof
Owing to Hardy’s inequality [18, p. 28], any function \(u_0\in \{v\in H^1_0(\Omega ): v_j\in H^2(\Omega _j), \forall j=1,\cdots , J\}\) admits the splitting
with \(u_w\in \mathcal {V}^2_\varepsilon (\Omega )\) for all \(\varepsilon >0\). This implies that \(u_0\in \hat{D}_1\). Since \(L^2(\Omega )\hookrightarrow L^2_\gamma (\Omega )\), we will have \(A_1^{-1} f\in C([0,T], D_1)\) and we conclude owing to the continuity of \(A_1^{-1} A_2\) from \(\hat{D}_1\) into itself. \(\square \)
In the framework of this corollary, a solution \(u\in C^1([0,T]; \hat{D}_1)\) is found. Therefore our convergence results will be guaranteed if we show (3.19) with \(q(1)=1\) and refined meshes but for any \(\varphi \in \hat{D}_1\). For that purpose, write \(\varphi \in \hat{D}_1\) into
with \(\varphi _w\in D_1\) and real coefficients \(c_s\in \mathbb {R}\). For the first term, by [36] or [18, Thm 8.4.1.6] we have
where \(I_h\) is the Lagrange interpolation operator. For the second term, as \(\eta _s\) belongs to \(H^2(\Omega )\), a standard interpolation estimate yields
In conclusion the function \(I_h\varphi \) satisfies
which proves (3.19) with \(q(1)=1\).
4.3 Numerical results
To validate our theoretical results, we propose different test examples. First in (4.2) we take \(L_1=I-\Delta \) and \(L_2=-\Delta \) (\(\Delta \) being the Laplace operator) in convex and non-convex polygons with an explicit solution and compute the different rates of convergence. Then we will consider a semi-linear equation for which the exact solution is unknown, hence we compute experimental convergence rates. In all cases, we compute two rates of convergence of the error (in the \(H^{1}_{0}(\Omega )\) norm): one in space and another one in time. Namely, for the first (resp. second) one, we chose \(\Delta t\) (resp. h) small enough with respect to h (resp. \(\Delta t\) ) so that the error due to the time (resp. space) discretization is neglectible; and then let vary the parameter h (resp. \(\Delta t\)) from a rough value to finer ones.
In the whole subsection, for a sequence of functions \(U_n\in H^1(\Omega ), 0\le n\le N\), we set
4.3.1 The smooth case
On the unit square \((0,1)^2\subset \mathbb {R}^2\), we take the exact solution
the right-hand side f being computed accordingly. In such a case, we present the numerical tests for the Euler scheme (4.7), where \(V_h\) is based on \(\mathbb {P}_1\) elements. The approximated solution obtained by this scheme is illustrated in Fig. 1 for different values of t with the choice \(\Delta t=h=0.1.\)
The rate of converge of the error in space (resp. time) is presented in Table 1 (resp. 2) with \(\Delta t=0.001\) (resp. \(h=1/160\)). There we can see a rate of convergence of 1, that is in accordance with (3.33).
We now present the numerical results relative to the Runge–Kutta scheme (4.8), where \(V_h\) is based on \(\mathbb {P}_2\) elements. In this case, as exact solution, we take
that then belongs to \(C([0,T];D(A_1^{\frac{3}{2}})\).
From Tables 3 and 4, we see that the convergence rate is 2 in space and 2 in time, as expected from (3.33).
4.3.2 The nonsmooth case
In order to illustrate the results from Sect. 4.2, we have decided to take the domain \(\Omega =(-1,1)\times (0,1)\subset \mathbb {R}^{2}\), the operator \(L_2=-\Delta \), while the operator \(L_1=-\) div \(a\nabla \), with a piecewise constant, namely
where \(\epsilon \) is a positive parameter, we have set
and, as usual, \((r,\theta )\) are the polar coordinates of (x, y) centred at the origin.
In that case if \(\epsilon \ne 1\), the operator \(L_1\) with Dirichlet boundary conditions has a singularity at (0, 0) given by (see [30])
with \(\lambda =\frac{4}{\pi }\arcsin \left( \sqrt{\frac{1}{\epsilon +1}}\right) \).
Consequently we take
that is seen as the exact solution of
where \(h\in \tilde{D}_0\) is the jump of \(\frac{\partial u}{\partial n}\) along the edge \(e_1=\bar{\Omega }_1\cap \bar{\Omega }_2\) and the edge \(e_2=\bar{\Omega }_2\cap \bar{\Omega }_3\). This means that u is solution of
We then have approximated this problem by the Euler scheme (4.7), where \(V_h\) is based on \(\mathbb {P}_1\) elements on either uniform meshes or refined (near 0) ones with the choice \(\epsilon =3\) that yields \(\lambda =2/3\). The rate of converge of the error in space is presented in Table 5 for uniform and refined meshes (see Fig. 2 for \(h=0.2\)) with \(\Delta t=0.0001\) and a final time \(T=0.1\). There we can see a rate of convergence of 2 / 3 (resp. 1) for uniform (refined) meshes, as expected. Here for shortness, we do not present the rate of converge of the error in time since we are interested in the influence of the space singularities.
Uniform (left) and refined (right) meshes
4.3.3 A semi-linear equation
Here we consider problem (4.2) on the unit square \(\Omega =(0,1)^2\) and zero initial datum with
that is clearly continously differentiable from \([0,T]\times H^1_0(\Omega )\) into \(\mathbb {R}\). In such a case, the exact solution is unknown, hence we shall compute the experimental rates of convergence using succesive solutions: the experimental space convergence rate is computed by
where \(U_{n,2h}\) and \(U_{n,h/2}\) are the fully discrete solutions for the meshes 2h and h / 2 respectively and \(\Delta t\) small enough. Similarly, the experimental time convergence rate is computed by
where \(U^{2\Delta t}_{n,h}\) and \(U^{\Delta t/2}_{n,h}\) are the fully discrete solutions for the time steps \(2\Delta t\) and \(\Delta t/2,\) respectively and h small enough.
The fully discrete solution \(U_{n,h}\) obtained by Euler’s scheme with \( \Delta t=0.1\) and \(h=0.1\) and \(\mathbb {P}_1\) el
Figure 3 shows the fully discrete solution \(U_{n,h}\) obtained by Euler’s scheme and \(\mathbb {P}_1\) elements at final time \(T=1\) with \(\Delta t=0.1\) and \(h=0.1.\) In that case, the experimental time (resp. space) convergence rate is presented in Table 6 (resp. 7), where an order one is detected, as theoretically expected. Additionnally, the experimental time (resp. space) convergence rate is presented in Table 8 (resp. 9) using \(\mathbb {P}_2\) elements, where, as theoretically expected, an order one in time and two in space are observed.
5 Applications to quasi-linear equations
5.1 Non autonomous equations
Here we concentrate on the non autonomous case, namely we suppose that the operators \(A_1(t,u)\) and \(A_2(t,u)\) depend only on the time variable t, but still corresponds to second order differential operators. More precisely, in a bounded domain \(\Omega \) of \(\mathbb {R}^{d},d\ge 1\) with a Lipschitz boundary, for \(i=1,2,\)\(L_i\) is a differential operator of order two of the form
where \(a^{(i)}_{k,\ell }=a^{(i)}_{\ell , k}\in C([0,T];C^{0,1}(\bar{\Omega }))\cap C^{0,\gamma }([0,T];L^{\infty }(\Omega ))\), \(b^{(i)}_{k}, c^{(i)}\in C^{0,\gamma }([0,T];L^\infty (\Omega ))\), for some \(\gamma \in (0,1]\). Furthermore, \(L_1\) is supposed to be uniformly elliptic, namely there exists \(\alpha _*>0\) such that
In this case, the bilinear form \(a_i(t;\cdot ,\cdot ), i=1,2,\) is independent of u and is defined by
and for all \(t\in [0,T],\) the operator \(A_i(t)\) defined by
is continous from \(H^{1}_0(\Omega )\) into \(H^{-1}(\Omega ).\) Finally if we suppose that \(a_1\) is uniformly coercive in \(H^{1}_0(\Omega )\), then the assumptions \(\mathbf {H_1}\)–\(\mathbf {H_3}\) will be satisfied. As in Sect. 4.1, one can show that \(D(A_1(t))=H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\), for all \(t\in [0,T]\) if the boundary of \(\Omega \) is \(C^{1,1}\) or if \(\Omega \) is convex. Therefore, under this additional hypothesis, the assumptions \(\mathbf {H_7}\) and \(\mathbf {H_8}\) with \(D_1=H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\) and \(\tilde{D}_{0}=L^2(\Omega )\) will be satisfied if \(V_h\) is defined by (4.6) with \(p=1\).
Finally, under the assumptions \(\mathbf {H_4}\) to \(\mathbf {H_6}\) on f, the next problem
is well-posed and can be approxiamted by the fully Euler discrete scheme (see (3.16))
We now illustrate this theory by chosing in (5.1), \(\Omega =(0,1)^2\), \(L_1(t)=I-(1+t)\Delta \) and \(L_2(t)=-(1+t)\Delta .\) Clearly the bilinear forms \((a_i(t;\cdot ,\cdot ))_{t\in [0,T]}\) satisfy the previous assumptions, in particular we directly see that
We start with a linear problem by taking the exact solution
As before, we present the time (resp. space) convergence rate in Table 10 with \(h=0.003125\) (resp. Table 11 with \(\Delta t=0.001\)), where again order one is obtained.
We go on with a semi-linear equation by taking the source term f(t, u) defined by (4.14) and a zero initial datum \(u(0)=0.\) In Fig. 4, we can see the fully discrete solution (by Euler’s scheme) \(U_{n,h}\) at final time \(T=1\) with \(\Delta t=h=0.1.\) The experimental time (resp. space) convergence rate is presented in Table 12 with \(h=\frac{1}{160}\) (resp. 13 with \(\Delta t=\frac{1}{4000}\)), where again an order one is detected.
The fully discrete solution \({U}_{n,h}\) at final time \(T=1\) obtained by Euler’s scheme with \(\Delta t=h=0.1\)
5.2 Quasi-linear cases
5.2.1 An example in dimension 1
Here we consider a quasilinear problem
where \(\rho _1, \rho _2:(0,1)\times \mathbb {R}\longmapsto \mathbb {R}\) are two continuous functions satisfying
-
there exist two positive constants \(\beta , M\) such that
$$\begin{aligned} \beta \le \rho _1(x,u)\le M, \;\;\text { and }\;\;|\rho _2(x,u)|\le M, \forall (x,u)\in \bar{\Omega }\times \mathbb {R}, \end{aligned}$$(5.3) -
the function \(\rho _i, i=1,2\), is globaly Lipschitz, i. e., there exists a constant \(L>0\) such that
$$\begin{aligned} |\rho _i(x,u)-\rho _i(x,\tilde{u})|\le L|u-\tilde{u}|,\;\;\forall (x, u,\tilde{u})\in \bar{\Omega }\times \mathbb {R}^{2}. \end{aligned}$$(5.4)
With these assumptions, the bilinear forms \(a_i(u;\cdot ,\cdot ), i=1,2,\) defined on \(H^{1}_0(\Omega )\times H^{1}_0(\Omega )\) as
satisfy the assumptions \(\mathbf {H_1}\)–\(\mathbf {H_3}\), this last property following from the Sobolev embedding theorem yielding \(H^1(0,1)\hookrightarrow C([0,1])\).
Here we discretize problem (5.2) by explicit Euler’s scheme using the finite element space
based on a uniform subdivision \(x_i=ih\), \(0\le i\le N\), with \(h=\frac{1}{N}\) and \(N\in \mathbb {N}^*\). For the numerical illustrations, we take the source term f(t, u) defined by (4.14), a zero initial datum \(u(0)=0\) and
As before the experimental time (resp. space) convergence rate is 1, as seen in Table 14 with \(h=\frac{1}{320}\) (resp. 15 with \(\Delta t=\frac{1}{1000}\)).
5.2.2 An example in dimension 2
On the unit square \(\Omega =(0,1)^2\) of \(\mathbb {R}^2,\) we consider the problem
where \(\frac{\partial u}{\partial n}\) denote the outward normal derivative of u on \(\partial \Omega ,\)\(\rho _1\) and \(\rho _2\) are two functions in \(C^2(\bar{\Omega }\times \mathbb {R},\mathbb {R})\) that fulfil the assumptions (5.3)–(5.4) with second order partial derivatives uniformly bounded in x and u.
For all \(u\in H^{2}_0(\Omega ),\) and \(i=1\) or 2, we define the bilinear form \(a_i(u;\cdot ,\cdot )\) by
that immediately satisfy the assumptions \(\mathbf {H_1}\) and \(\mathbf {H_3}\), due to the embedding \(H^2(\Omega )\hookrightarrow C(\bar{\Omega })\) (consequence of the Sobolev embedding theorem). To check that \(\mathbf {H_2}\) holds, due to (5.3) we first notice that
Secondly as the Laplace operator is an isomorphism from \(H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\) into \(L^2(\Omega )\), we have
As \(H^{2}_{0}(\Omega )\) is included into \(H^{2}(\Omega )\cap H^{1}_{0}(\Omega )\), these two estimates show that \(\mathbf {H_2}\) holds.
In order to check \(\mathbf {H_7}\) and \(\mathbf {H_8}\), we use the next result.
Lemma 5.1
For all \(u\in H^{2}_0(\Omega ),\) we have
Proof
For a fixed \(u\in H^{2}_0(\Omega ),\) we recall that
Hence we see that \(v\in H^{2}_0(\Omega )\) belongs to \(D(A_1(u))\) if and only if there exists \(f\in L^2(\Omega )\) such that
in the distributional sense (here and below, for shortness we write \(\rho (x,u)\) for \(\rho _1(x,u)\)). By Leibniz’s product rule, we get equivalently
The difficulty is that this right-hand side is not automatically in \(L^2(\Omega )\), hence we will use a bootstrap argument. First by the chain rule, we notice that
for any \(i,j\in \{1,2\}\). Note that these identities directly imply that
With such identities, for \(i=1\) or 2, we also see that
and that
where we have set
In a first step, for \(i=1\) or 2, we show that
According to the identity (5.12), it suffices to show that each term of its right-hand side belongs to \(H^{-s}(\Omega ),\) for all \(s\in (1,2).\) For that purpose, we notice that \( \partial _i \Delta v\in H^{-1}(\Omega )\), and \(\partial _i u\in H^1(\Omega )\). By the regularity assumptions on \(\rho \), we see that
As Theorem 1.4.4.2 of [18] implies that the product
If \(u\in H^1(\Omega )\) and \(v\in H^{-1}(\Omega )\), we conclude that (5.14) holds. Similarly for \(j=1,2\) or 3, we show that
By the boundedness of \(\Delta \rho \) and the regularity \(v\in H^2(\Omega )\), we directly get \(T_1(x,u) \Delta v\in L^2(\Omega )\), hence (5.17) for \(j=1\). Now by the Sobolev embedding Theorem, \(H^1(\Omega )\hookrightarrow L^p(\Omega )\), for all \(p\ge 1\), hence by Hölder’s inequality, we get
since the condition \(1=\frac{2}{p}+\frac{1}{2}\) holds if \(p=4\). As \(\partial ^2_u \rho (x,u)\) is bounded, we deduce that \(T_2(x,u) \Delta v\in L^1(\Omega )\), which implies (5.17) for \(j=2\), because \(H^s_0(\Omega )\hookrightarrow C (\bar{\Omega })\), owing to the Sobolev embedding theorem. Finally, the regularities \(u,v\in H^2(\Omega )\) simply guarantee that \(\Delta u \Delta v\in L^1(\Omega )\) and hence \(T_3(x,u) \Delta v\in L^1(\Omega )\), and we conclude as for \(j=2\).
At this stage, by (5.14) and (5.17), we deduce that \(\tilde{f}\) (defined in (5.8)) belongs to \(H^{-s}(\Omega ),\) for all \(s\in (1,2).\) With the regularity property (5.11) and Theorem 1.4.4.2 of [18], we conclude that
Owing to Theorem 2 of [6] and Corollary 5.12 of [13], we deduce that
or equivalently
This extra regularity allows to show that
for \(i=1\) or 2 and
for \(j=1,2\) or 3. For the first assertion, by the regularity \( \partial _i \Delta v\in H^{-\varepsilon }(\Omega )\), the properties (5.15) and Theorem 1.4.4.2 of [18], we conclude that (5.18) holds for \(\varepsilon '>\varepsilon \). For the second assertion, by the boundedness of second derivatives of \(\rho _1\), we first notice that
hence (5.19) for \(j=1\). For \(j=2\) we remark that \((\partial _1 u)^2+(\partial _2 u)^2\) belongs to \(L^2(\Omega )\) due to the embedding \(H^1(\Omega )\hookrightarrow L^p(\Omega )\), for all \(p\ge 1\), and Hölder’s inequality. Since we directly get that \(\Delta u\in L^2(\Omega )\), we conclude that
Again Theorem 1.4.4.2 of [18] leads to (5.19) for \(\varepsilon '>\varepsilon \), and \(j=2,3\).
By its definition, we deduce that \(\tilde{f}\) belongs to \(H^{-\varepsilon ' }(\Omega ),\) for all \(\varepsilon ' \in (0,1),\) and by (5.11) and Theorem 1.4.4.2 of [18] we get
By Theorem 2 of [6] and Corollary 5.12 of [13], we deduce that
This regularity implies that \(\partial _i \Delta v\in H^{1-\varepsilon }(\Omega ),\) for all \(\varepsilon \in (0,1)\) hence \(\partial _i \Delta v\in L^{p}(\Omega ),\) for all \(p>1\), which allows to show that
More simply as \( \Delta v\) belongs to \(H^{2-\varepsilon }(\Omega ),\) for all \(\varepsilon \in (0,1)\), it is bounded in \(\Omega \) and consequently
This leads to the final property
and by Theorem 2 of [6], we conclude that
We have thus shown that \( D(A_1(u))\hookrightarrow H^4(\Omega )\cap H^{2}_0(\Omega )\). As the converse embedding is direct the proof is complete. \(\square \)
With the help of this Lemma, the assumption \(\mathbf {H_7}\) holds with the choice \(D_1=H^4(\Omega )\cap H^{2}_0(\Omega )\) and \(\tilde{D}_{0}=L^2(\Omega ).\)
Since the variational space is included into \(H^2(\Omega ),\) the use of continuous FEM is not appropriate, hence we shall use HCT elements (Hsieh-Cough-Tocher) described in [10] for instance. Such elements are macro-elements (see Fig. 5) since each triangle K is subdivided into three sub-triangles \(K_i, i=1,2,3\), namely we define
and then
For more details, we refer to [10, p. 341].
The HCT element
As Theorem 6.1.6 of [10] implies that
the assumption \(\mathbf {H_8}\) holds with \(q(1)=2\).
The fully discrete explicit Euler’s scheme of problem (5.5) is therefore given by: find \(U_{n+1,h}\in V_h\) solution of
The fully discrete solution: Left u, middle \(\partial _1 u\), right \(\partial _2 u\), with \(\Delta t=h=0.1\) at final time \(T=1\)
We finally illustrate this case by chosing \(\rho _1(x,u)=\frac{1}{2}+\frac{u^2}{1+u^2},\)\(\rho _2(x,u)=\frac{u^2}{1+u^2}\), the source term f(t, u) defined by (4.14) and a zero initial datum \(u(0)=0.\) In Fig. 6, we can see the fully discrete solution \(U_{n,h}\) and its gradient at final time \(T=1\) with \(\Delta t=h=0.1.\) The experimental time (resp. space) convergence rate is presented in Table 16 with \(h=0.00625\) (resp. 17 with \(\Delta t=\frac{1}{1000}\)), where an order one in time and two in space is obtained, as expected from (3.33).
Note that all our numerical tests are performed with the help of the software freefem++ [1].
References
Arnold, D.N., Douglas Jr., J., Thomée, V.: Superconvergence of a finite element approximation to the solution of a Sobolev equation in a single space variable. Math. Comput. 36(153), 53–63 (1981)
Avilez-Valente, P., Seabra-Santos, F.J.: A Petrov-Galerkin finite element scheme for the regularized long wave equation. Comput. Mech. 34(4), 256–270 (2004)
Barenblatt, G.I., Entov, V.M., Ryzhik, V.M.: Theory of Fluid Flow Through Natural Rocks. Kluwer, Dordrecht (1990)
Bernardi, C.: Optimal finite-element interpolation on curved domains. SIAM J. Numer. Anal. 26(5), 1212–1240 (1989)
Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2(4), 556–581 (1980)
Carroll, R.W., Showalter, R .E.: Singular and Degenerate Cauchy problems, vol. 127. Academic, New York (1976). Mathematics in Science and Engineering
Chatzipantelidis, P.: Explicit multistep methods for nonstiff partial differential equations. Appl. Numer. Math. 27(1), 13–31 (1998)
Chen, P.J., Gurtin, M.E.: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys. 19, 614–627 (1968)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
Coleman, B.D., Noll, W.: An approximation theorem for functionals, with applications in continuum mechanics. Arch. Ratl. Mech. Anal. 6(355–370), 1960 (1960)
Cuesta, C., van Duijn, C.J., Hulshof, J.: Infiltration in porous media with dynamic capillary pressure: travelling waves. Eur. J. Appl. Math. 11(4), 381–397 (2000)
Dauge, M.: Elliptic Boundary Value Problems on Corner Domains—Smoothness and Asymptotics of solutions, Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)
Dogan, A.: Numerical solution of regularized long wave equation using Petrov–Galerkin method. Commun. Numer. Methods Eng. 17(7), 485–494 (2001)
Ewing, R.E.: Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations. SIAM J. Numer. Anal. 15(6), 1125–1150 (1978)
Gajewski, H., Zacharias, K.: Über eine weitere Klasse nichtlinearer Differentialgleichungen im Hilbert-Raum. Math. Nachr. 57, 127–140 (1973)
Gao, F., Wang, X.: A modified weak Galerkin finite element method for Sobolev equation. J. Comput. Math. 33(3), 307–322 (2015)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)
Hairer, E., Nø rsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I, Volume 8 of Springer Series in Computational Mathematics. Springer, Berlin (1993)
Hassanizadeh, S.M., Gray, W.G.: Thermodynamic basis of capillary pressure in porous media. Water Resour. Res. 29, 858–879 (1993)
Hell, T., Ostermann, A., Sandbichler, M.: Modification of dimension-splitting methods–overcoming the order reduction due to corner singularities. IMA J. Numer. Anal. 35(3), 1078–1091 (2015)
Kadlec, J.: The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain. Czech. Math. J. 14(89), 386–393 (1964)
Kato, T.: Quasi-linear equations of evolution, with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Math., vol. 448, pp. 25–70. Springer, Berlin (1975)
Kellogg, R.B.: Singularities in interface problems. In: Hubbard, B. (ed.) Numerical Solution of Partial Differential Equations II, pp. 351–400. Academic, New York (1971)
Kellogg, R.B.: On the Poisson equation with intersecting interfaces. Appl. Anal. 4, 101–129 (1975)
Lemrabet, K.: Régularité de la solution d’un problème de transmission. J. Math. Pures Appl. 56, 1–38 (1977)
Lions, J.-L.: Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften, Bd. 111. Springer, Berlin (1961)
Liu, T., Lin, Y.-P., Rao, M., Cannon, J.R.: Finite element methods for Sobolev equations. J. Comput. Math. 20(6), 627–642 (2002)
Nicaise, S.: Polygonal interface problems, Methoden und Verfahren der Mathematischen Physik, vol. 39. Verlag Peter D. Lang, Frankfurt am Main (1993)
Nicaise, S., Sändig, A.-M.: General interface problems. I, II. Math. Methods Appl. Sci. 17(6):395–429, 431–450 (1994)
Ohm, M.R., Lee, H.Y.: \(L^2\)-error analysis of fully discrete discontinuous Galerkin approximations for nonlinear Sobolev equations. Bull. Korean Math. Soc. 48(5), 897–915 (2011)
Ohm, M.R., Lee, H.Y., Shin, J.Y.: \(L^2\)-error estimates of the extrapolated Crank-Nicolson discontinuous Galerkin approximations for nonlinear Sobolev equations. J. Inequal. Appl., pages Art. ID 895187, 17 (2010)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Ptashnyk, M.: Pseudoparabolic equations with convection. IMA J. Appl. Math. 72(6), 912–922 (2007)
Quarteroni, A.: Fourier spectral methods for pseudoparabolic equations. SIAM J. Numer. Anal. 24(2), 323–335 (1987)
Raugel, G.: Résolution numérique par une méthode d’éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. A-B 286(18), A791–A794 (1978)
Schwartz, L.: Mathematics for the Physical Sciences. Hermann; Addison-Wesley Publishing Co., Paris, Reading (1966)
Showalter, R.E.: The Sobolev equation. I. Appl. Anal. 5(1), 15–22 (1975)
Showalter, R.E.: The Sobolev equation. II. Appl. Anal. 5(2), 81–99 (1975)
Showalter, R.E.: Hilbert Space Methods for Partial Differential Equations, Monographs and Studies in Mathematics, vol. 1. Pitman, London (1977)
Ting, T.W.: Certain non-steady flows of second-order fluids. Arch. Ratl. Mech. Anal. 14, 1–26 (1963)
Ting, T.W.: A cooling process according to two-temperature theory of heat conduction. J. Math. Anal. Appl. 45, 23–31 (1974)
Wahlbin, L.: Error estimates for a Galerkin method for a class of model equations for long waves. Numer. Math. 23, 289–303 (1975)
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Bekkouche, F., Chikouche, W. & Nicaise, S. Fully discrete approximation of general nonlinear Sobolev equations. Afr. Mat. 30, 53–90 (2019). https://doi.org/10.1007/s13370-018-0626-9
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DOI: https://doi.org/10.1007/s13370-018-0626-9