Abstract
A mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems is presented, and convergence with optimal error estimates is proved in the framework of \(L^2\)-theory. The introduced scheme maintains the advantages of the Lagrange–Galerkin method, i.e., CFL-free robustness for convection-dominated problems and a symmetric and positive coefficient matrix resulting from the discretization. In addition, the scheme conserves the mass on the discrete level if the involved integrals are computed exactly. Unconditional stability and error estimates of second order in time are proved by employing two new key lemmas on the truncation error of the material derivative in conservative form and on a discrete Gronwall inequality for multistep methods. The mass-preserving property is achieved by the Jacobian multiplication technique introduced by Rui and Tabata in 2010, and the accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics originally introduced by Ewing and Russel in 1981. For the first time step, the mass-preserving scheme of first order in time by Rui and Tabata in 2010 is employed, which is efficient and does not cause any loss of convergence order in the \(\ell ^\infty (L^2)\)- and \(\ell ^2(H^1_0)\)-norms. For the time increment \(\Delta t\), the mesh size h and a conforming finite element space of polynomial degree \(k \in {\mathbb {N}}\), the convergence order is of \(O(\Delta t^2 + h^k)\) in the \(\ell ^\infty (L^2)\cap \ell ^2(H^1_0)\)-norm and of \(O(\Delta t^2 + h^{k+1})\) in the \(\ell ^\infty (L^2)\)-norm if the duality argument can be employed. Error estimates of \(O(\Delta t^{3/2}+h^k)\) in discrete versions of the \(L^\infty (H^1_0)\)- and \(H^1(L^2)\)-norm are additionally proved. Numerical results confirm the theoretical convergence orders in one, two and three dimensions.
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1 Introduction
The convection-diffusion equation is one of the important equations in flow problems, as it is considered a simplification of the Navier–Stokes equations. To deal with the equation especially in convection-dominant cases, nowadays, many finite element schemes have been proposed and analyzed, e.g., upwind methods [2, 9, 10, 21, 22, 38], characteristics(-based) methods [4,5,6, 11, 13,14,15,16,17, 30,31,32, 34, 35, 39] and so on. The Lagrange–Galerkin method (also called characteristic(-curve) finite element method or Galerkin-characteristics method) belongs to the latter group and is a finite element method based on the method of characteristics, where the idea is to consider the trajectory of a fluid particle and discretize the material derivative along this trajectory. It is known that the Lagrange–Galerkin method has many advantages including robustness for convection-dominated problems without needing any stabilization parameters, symmetry of the resulting coefficient matrix, and no requirement of the so-called CFL condition, which enables the use of large time increments. Hence, the Lagrange–Galerkin method has also been applied to other equations, e.g., the Oseen/Navier–Stokes/viscoelastic/natural convection equations, cf. [3, 7, 8, 23,24,25,26,27,28,29, 37] and references therein.
Some Lagrange–Galerkin schemes of second order in time for convection-diffusion problems have already been proposed, including single step methods [4, 5, 34] and multistep methods [6, 16]. However, in general, the mass-preserving property is often not satisfied by Lagrange–Galerkin methods. Recently mass-preserving Lagrange–Galerkin schemes for convection-diffusion problems in conservative form and hyperbolic conservation laws, i.e., pure convection problems in conservative form, with arbitrary orders in time and space have been proposed by Colera et al. [13, 14] but error estimates are not yet given. About a decade ago, Rui and Tabata [32] has proposed a mass-preserving Lagrange–Galerkin scheme of first order in time for convection-diffusion problems by a Jacobian multiplication technique and proved error estimates of first order in time. To the best of our knowledge, however, there are no Lagrange–Galerkin schemes of second order in time having both, a mass-preserving property and error estimates.
In this paper, we propose a Lagrange–Galerkin scheme of second order in time for convection-diffusion problems and prove its mass-preserving property and error estimates. Stability and convergence with optimal error estimates are proved in the framework of \(L^2\)-theory. We devise the scheme based on two ideas; one is the multistep (two-step) Galerkin method along characteristics by Ewing and Russel [16], and the other one is the Jacobian multiplication technique by Rui and Tabata [35]. To find the numerical solution at time step n, we employ two Jacobians for the time steps \(n-1\) and \(n-2\). The Jacobians are of the forms, \(1 - \Delta t (\nabla \cdot u^n) + O(\Delta t^2)\) and \(1 - 2\Delta t (\nabla \cdot u^n) + O(\Delta t^2)\), respectively, where \(\Delta t\) is a time increment and \(u^n\) is the velocity at time step n. For this reason it is not obvious that our scheme is of second order in time and that the mass-preserving property is satisfied. We, therefore, prove these properties in this paper. As two-step methods require solutions at two prior time steps, we propose to employ the mass-preserving Lagrange–Galerkin scheme of first order in time by Rui and Tabata [35] for the first time step. This construction is efficient and does not cause any loss of convergence order in the \(\ell ^\infty (L^2)\)- and \(\ell ^2(H^1_0)\)-norms.
The main results for our scheme including the construction of the solution at the first time step are as follows. (i) The mass-preserving property is proved, cf. Theorem 1. (ii) Stability in \(\ell ^\infty (L^2)\cap \ell ^2(H^1_0)\) and \(\ell ^\infty (H^1_0)\) is proved, cf. Theorem 2. (iii) An error estimate of \(O(\Delta t^2 + h^k)\) in the \(\ell ^\infty (L^2)\cap \ell ^2(H^1_0)\)-norm is proved, where h is the mesh size in space and \(k \in {\mathbb {N}}\) is the polynomial degree of a conforming finite element space for the numerical solution, cf. Theorem 3-(i). (iv) An error estimate of \(O(\Delta t^2 + h^{k+1})\) in the \(\ell ^\infty (L^2)\)-norm is proved under the assumption that the duality argument can be employed, cf. Theorem 3-(ii). Furthermore, in Theorem 3-(i), we prove an error estimate of \(O(\Delta t^{3/2} + h^k)\) in a discrete version of the \(L^\infty (H^1_0)\cap H^1(L^2)\)-norm. Although the convergence order in the \(L^\infty (H^1_0)\cap H^1(L^2)\)-norm is slightly reduced to \(\Delta t^{3/2}\) due to the construction of the solution at the first time step, it is still higher than first order. When we consider an application of the scheme to the Navier–Stokes equations, the further analysis will be useful for the estimate of the pressure.
Here, we make two further remarks. (i) In real computations our scheme is only approximately mass conservative, since numerical integration is in general required to compute the integrals occuring in the scheme. This introduces an approximation error in the total mass of the discrete solution. In this paper, in place of mass-conservative, which we only use if no mass is lost (in the discrete case up to machine precison), we employ the term mass-preserving to refer to schemes that are mass-conservative if the involved integrals are computed exactly. (ii) While there are \(\ell ^\infty (L^2)\)-error estimates for single-step Lagrange–Galerkin methods (including space-time versions) for convection-diffusion problems that are independent of the viscosity constant, cf., e.g., [11, 35, 39], the error estimates in this paper are dependent on the viscosity constant. This is caused by an estimate of the discrete material derivative using the two-step backward differentiation formula in combination with the discrete Gronwall’s inequality for the two-step method and to the best of our knowledge no viscosity-independent error estimates for multi-step Lagrange–Galerkin methods exist. Furthermore, in applications to the Navier–Stokes equations, viscosity-dependent error estimates are usually obtained even for single-step Lagrange–Galerkin methods due to the nonlinearity.
This paper is organized as follows. Our mass-preserving two-step Lagrange–Galerkin scheme for convection-diffusion problems is presented in Sect. 2. The main results on the mass-preserving property, the stability, and the convergence with optimal error estimates are stated in Sect. 3, and they are proved in Sect. 4. The theoretical convergence orders are numerically confirmed by one-, two- and three-dimensional numerical experiments in Sect. 5. The conclusions are given in Sect. 6. In the Appendix three lemmas used in Sect. 4 are proved.
2 A Lagrange–Galerkin Scheme
The function spaces and the notations used throughout the paper are as follows. Let \(\Omega \) be a bounded domain in \({\mathbb {R}}^d\) for \(d = 1, 2\) or 3, \(\Gamma :=\partial \Omega \) the boundary of \(\Omega \), and T a positive constant. For \(m \in {\mathbb {N}}\cup \{0\}\) and \(p\in [1,\infty ]\), we use the Sobolev spaces \(W^{m,p}(\Omega )\), \(W^{1,\infty }_0(\Omega )\), \(H^m(\Omega ) \, (=W^{m,2}(\Omega ))\) and \(H^1_0(\Omega )\). For any normed space S with norm \(\Vert \cdot \Vert _S\), we define function spaces \(H^m(0,T; S)\) and C([0, T]; S) consisting of S-valued functions in \(H^m(0,T)\) and C([0, T]), respectively. We use the same notation \((\cdot , \cdot )\) to represent the \(L^2(\Omega )\) inner product for scalar- and vector-valued functions. The norm on \(L^2(\Omega )\) is simply denoted by \(\Vert \cdot \Vert \), i.e., \(\Vert \cdot \Vert :=\Vert \cdot \Vert _{L^2(\Omega )}\). The dual pairing between S and the dual space \(S^\prime \) is denoted by \(\langle \cdot , \cdot \rangle \). The notation \(\Vert \cdot \Vert \) is employed not only for scalar-valued functions but also for vector-valued ones. For \(t_0\) and \(t_1\in {\mathbb {R}}\) \((t_0 < t_1)\), we introduce the function space
with the norm
and set \(Z^m :=Z^m(0, T)\). We often omit [0, T], \(\Omega \), and the superscript d if there is no confusion, e.g., we shall write \(C(L^\infty )\) in place of \(C([0,T]; L^\infty (\Omega )^d)\). We denote by c and \(c(a_1, a_2, \ldots )\) a generic positive constant and a positive constant dependent on \(a_1, a_2, \ldots \), respectively, and introduce the following constants, for \(i=0,1\),
We consider a convection-diffusion problem; find \(\phi :\Omega \times (0, T)\rightarrow {\mathbb {R}}\) such that
where \(u:\Omega \times (0, T)\rightarrow {\mathbb {R}}^d\), \(f:\Omega \times (0, T)\rightarrow {\mathbb {R}}\), \(g: \Gamma \times (0, T) \rightarrow {\mathbb {R}}\) and \(\phi ^0:\Omega \rightarrow {\mathbb {R}}\) are given functions, \(n: \partial \Omega \rightarrow {\mathbb {R}}^d\) is the outward unit normal vector, \(\nu \in (0, \nu _0]\) is a viscosity constant, and \(\nu _0 (>0)\) is an upper bound of \(\nu \). Since we are interested in problems with a small \(\nu \), i.e., convection-dominated problems, we assume without loss of generality \(\nu _0 = 1\) in this paper.
Let \(\Psi :=H^1(\Omega )\). A weak formulation to problem (1) is to find \(\{\phi (t) = \phi (\cdot ,t) \in \Psi ;\ t \in (0, T)\}\) such that, for \(t\in (0, T)\),
with \(\phi (0) = \phi ^0\), where \(a_0(\cdot ,\cdot )\) and \(a_1(\cdot \,,\cdot ) = a_1(\cdot \,,\cdot \,;u)\) are bilinear forms defined by
and \(F(t)\in \Psi ^\prime \), \(t\in (0,T)\), is a functional defined by
for \(f(t) = f(\cdot ,t) \in L^2(\Omega )\) and \(g(t) = g(\cdot ,t) \in L^2(\Gamma )\).
Let us assume \(f\in L^2(0,T;L^2(\Omega ))\) and \(g\in L^2(0,T;L^2(\Gamma ))\). Substituting \(1\in \Psi \) into \(\psi \) in (2) and integrating over (0, t), one can easily obtain the so-called mass-balance identity, i.e., for \(t\in (0, T)\),
which is an important property of problem (1). This property is, therefore, desired to hold also on the discrete level. It is known that conventional Galerkin, streamline diffusion (SD) [18, 22], streamline upwind/Petrov–Galerkin (SUPG), and least square schemes [10, 20] satisfy a discrete version of (4). In [35], a characteristic finite element (Lagrange–Galerkin) scheme of first order in time satisfying a discrete version of (4) has been proposed and analyzed.
Let \(\Delta t >0\) be a time increment, \(t^n :=n\Delta t~(n\in {\mathbb {Z}})\), and \(N_T :=\lfloor T/\Delta t\rfloor \). For a function \(\rho \) defined in \(\Omega \times (0, T)\), \(\rho (\cdot , t^n)\) is simply denoted by \(\rho ^n\). Let \({\mathcal {T}}_h\) be a triangulation of \(\Omega \), and \(\Omega _h :=\mathrm{int} ( \bigcup _{K\in {\mathcal {T}}_h} K )\) the approximate domain, where h is the maximum mesh size of \({\mathcal {T}}_h\), i.e., \(h :=\max \{ h_K;\ K\in {\mathcal {T}}_h \}\) for \(h_K:=\mathrm {diam} (K)\) \((K\in {\mathcal {T}}_h)\). For the sake of simplicity, we assume that \(\Omega _h = \Omega \) throughout this paper. Let \(\Psi _h\) be a finite element space defined by
where \(P_k(K)\) is the space of polynomial functions of degree \(k \in {\mathbb {N}}\) on \(K \in {\mathcal {T}}_h\). For a velocity \(v: \Omega \rightarrow {\mathbb {R}}^d\), let \(X_1 (v,\Delta t): \Omega \rightarrow {\mathbb {R}}^d\) be the mapping defined by
which is called the upwind point of x with respect to the velocity v and the time increment \(\Delta t\). We define mappings \(X_1^n, {\tilde{X}}_h^n: \Omega \rightarrow {\mathbb {R}}^d\) and their Jacobians \(\gamma ^n, {\tilde{\gamma }}^n: \Omega \rightarrow {\mathbb {R}}\) by
The scheme proposed in [35] is to find at each time step \(\phi _h^n\in \Psi _h\) such that
By multiplication with the Jacobian \(\gamma ^n\) the mass of \(\phi _h^{n-1}\) is conserved after taking the composite with the mapping \(X_i^n\) and we call this “the Jacobian multiplication technique.” That is substituting \(1\in \Psi _h\) into \(\psi \) in (7) and using the identity
we obtain a discrete mass-balance identity, cf. [35] for detail.
Moreover, a multistep (two-step) Galerkin method along characteristics of second order in time [16] is well known; at each time step \(n \in \{ 2, \ldots , N_T\}\), find \(\phi _h^n\in \Psi _h\) such that
Scheme (8) is of second order in time but does not satisfy the mass-balance identity in general.
Combining the Jacobian multiplication technique (7) with the multistep (two-step) Galerkin method along characteristics (8), we obtain the Lagrange–Galerkin scheme proposed in this paper.
Let \(\phi _h^0 \in \Psi _h\) and \(F \in H^1(0,T; \Psi ^\prime )\) be given. We propose a mass-preserving two-step Lagrange–Galerkin scheme of second order in time; find \(\{\phi _h^n\in \Psi _h;\ n=1, \ldots , N_T\}\) such that, for \(n=1, \ldots , N_T\),
Since the Jacobians \(\gamma ^n\) and \(\tilde{\gamma ^n}\) are of the forms \(1 - \Delta t (\nabla \cdot u^n) + O(\Delta t^2)\) and \(1 - 2\Delta t (\nabla \cdot u^n) + O(\Delta t^2)\), respectively, it is not clear that the combined scheme is of second order in time and that the mass-balance identity is satisfied. These properties are therefore proved in this paper. In the following, we rewrite scheme (9) simply as
for \(n \in \{1,\ldots , N_T\}\), where, for a series \(\{\rho ^n\}_{n=0}^{N_T} (\subset \Psi )\), the function \({\mathcal {A}}_{\Delta t} \rho ^n: \Omega \rightarrow {\mathbb {R}}\) is defined by
Remark 1
(i) The first order scheme (9a) is employed in the first time step, since then the approximate solution \(\phi _h^1\) needed in (9b) with \(n=2\) is not yet available. This construction of \(\phi _h^1\) is efficient and has no adverse effect on the convergence order in the \(\ell ^\infty (L^2)\)-norm, cf. Theorem 3.
(ii) \(F \in H^1(0,T; \Psi ^\prime )\) implies that \(F \in C([0,T]; \Psi ^\prime )\) and \(\{F^n\}_{n=1}^{N_T} \subset \Psi ^\prime \).
3 Main Results
We start this section, by setting hypotheses for the velocity u and the time increment \(\Delta t\), and reviewing previous results.
Hypothesis 1
The function u satisfies \(u \in C([0,T]; W^{1,\infty }_0(\Omega )^d)\).
Hypothesis 2
The time increment \(\Delta t\) satisfies the condition \(\Delta t |u|_{C(W^{1,\infty })} \le 1/8\).
Proposition 1
[34, 40] (i) Under Hypothesis 1 and \(\Delta t |u|_{C(W^{1,\infty })} < 1/2\), it holds that \(X_1^n(\Omega )={\tilde{X}}_1^n(\Omega ) = \Omega \) for \(n=0, \ldots , N_T\).
(ii) Under Hypotheses 1 and 2, it holds that \(1/2 \le \gamma ^n, {\tilde{\gamma }}^n \le 3/2\) for \(n=0, \ldots , N_T\).
For \(n=0, \ldots , N_T\), let \({\mathcal {M}}_h^n\) be an approximate value of mass at \(t=t^n\) defined by
Remark 2
The value \({\mathcal {M}}_h^n\) is an approximation of \(\int _\Omega \phi ^n dx\) due to the relation \(\frac{3}{2}\phi ^n-\frac{1}{2}\phi ^{n-1} (= \phi ^{n+1/2} + O(\Delta t^2)) = \phi ^n + O(\Delta t)\) for any smooth function \(\phi \).
Theorem 1
(conservation of mass) Suppose that Hypotheses 1 and 2 hold true. Let \(\phi _h = \{\phi _h^n\}_{n=1}^T\) be a solution to scheme (9) for a given \(\phi _h^0\). Then, we have the following.
(i) It holds that, for \(n=0,\ldots , N_T\),
(ii) Assume \(f=0\) and \(g=0\) additionally. Then, for the solution to scheme (9), it holds that, for \(n=0,\ldots , N_T\),
Remark 3
The identity (10) is equivalent to
For a sequence \(\{\rho ^n\}_{n = 0}^{N_T}\), let \({\bar{D}}_{\Delta t}\) be the backward quotient operator defined by
where \({\bar{D}}_{\Delta t}^{(1)}\) and \({\bar{D}}_{\Delta t}^{(2)}\) are the first- and second-order backward difference quotient operators,
Let \(m \in \{ 0,\ldots ,N_T\}\) be an integer and Y be a normed space. When \(\{\rho ^n\}_{n = 0}^{N_T} \subset Y\), we define the norms \(\Vert \cdot \Vert _{\ell ^\infty _m(Y)}\) and \(\Vert \cdot \Vert _{\ell ^2_m(Y)}\) by
and let \(\Vert \rho \Vert _{\ell ^\infty (Y)} :=\Vert \rho \Vert _{\ell ^\infty _1(Y)}\) and \(\Vert \rho \Vert _{\ell ^2(Y)} :=\Vert \rho \Vert _{\ell ^2_1(Y)}\). When \(Y = L^2(\Omega )\), we omit \(\Omega \) from the norms, e.g., \(\Vert \rho \Vert _{\ell ^\infty (L^2)}\), and use the same notations \(\Vert \cdot \Vert _{\ell ^\infty _m (L^2)}\), \(\Vert \cdot \Vert _{\ell ^2_m (L^2)}\), \(\Vert \cdot \Vert _{\ell ^\infty (L^2)}\) and \(\Vert \cdot \Vert _{\ell ^2 (L^2)}\) also for a sequence of vector valued functions, e.g., \(\Vert \nabla \rho \Vert _{\ell ^\infty (L^2)} = \max _{n=1,\ldots ,N_T} \Vert \nabla \rho ^n\Vert _{L^2(\Omega )^d}\).
Proposition 2
(stability for a given \(\phi _h^1\)) Suppose that Hypothesis 1 holds true. Let \(F \in H^1(0,T; \Psi ^\prime )\) be given. Suppose that Hypothesis 2 holds true, and assume \(\Delta t \in {(0, 1)}\). For given functions \(\phi _h^0, \phi _h^1 \in \Psi _h\), let \(\{\phi _h^n\}_{n=2}^{N_T} \subset \Psi _h\) be the solution to scheme (9b). Then, we have the following:
(i) There exists a positive constant \(c_\dagger = c_\dagger ( \Vert u\Vert _{C(W^{1,\infty })}, T, 1/\nu )\) independent of h and \(\Delta t\) such that
(ii) Assume \(F \in H^1(0,T; L^2(\Omega ))\) additionally. Then, there exists a positive constant \({\bar{c}}_\dagger = {\bar{c}}_\dagger (\Vert u\Vert _{C(W^{1,\infty })}, T, 1/\nu )\) independent of h and \(\Delta t\) such that
Theorem 2
(stability) Suppose that Hypothesis 1 holds true. Let \(F \in H^1(0,T; \Psi ^\prime )\) be given. Suppose that Hypothesis 2 holds true, and assume \(\Delta t \in {(0, 1)}\). For a given function \(\phi _h^0 \in \Psi _h\), let \(\{\phi _h^n\}_{n=1}^{N_T} \subset \Psi _h\) be the solution to scheme (9). Then, we have the following:
(i) There exists a positive constant \(c_\ddagger =c_\ddagger (\Vert u\Vert _{C(W^{1,\infty })}, T, 1/\nu )\) independent of h and \(\Delta t\) such that
(ii) Assume \(F \in H^1(0,T; L^2(\Omega ))\) additionally. Then, there exists a positive constant \({\bar{c}}_\ddagger = {\bar{c}}_\ddagger (\Vert u\Vert _{C(W^{1,\infty })}, T, 1/\nu )\) independent of h and \(\Delta t\) such that
Remark 4
The assumption \(F \in H^1(0,T; L^2(\Omega ))\) in Theorem 2-(ii) implies \(g=0\), which is explicitly written in Corollary 1-(ii) below.
Corollary 1
(i) Suppose that the functional \(F \in H^1(0,T; \Psi ^\prime )\) is given by (3) with \(f\in H^1(0,T;L^2(\Omega ))\) and \(g\in H^1(0,T;L^2(\Gamma ))\), the stability estimate (14) in Theorem 2-(i) becomes
(ii) Suppose that the functional \(F \in H^1(0,T; \Psi ^\prime )\) is given by (3) with \(f\in H^1(0,T;L^2(\Omega ))\) and \(g=0\), the stability estimate (15) in Theorem 2-(ii) becomes
We present the convergence result of second order in time after stating regularity hypotheses for the solution to problem (2) given the polynomial degree \(k\in {\mathbb {N}}\) of the finite element space \(\Psi _h\) in Hypothesis 3 and for the solution of the Poisson problem in Hypothesis 4. Then we define the Poisson projection in Definition 1.
Hypothesis 3
The solution \(\phi \) to (2) satisfies \(\phi \in Z^3 \cap H^2(0,T; H^{k+1}(\Omega ))\).
Remark 5
We suppose \(H^2(0,T; H^{k+1}(\Omega ))\), since the regularity \(H^1(0,T; H^{k+1}(\Omega ))\) is not sufficient to get the convergence of second order in time, especially for the estimate of the solution at the first time step.
Hypothesis 4
The Poisson problem is regular on the domain \(\Omega \), i.e., for any \({\tilde{f}}\in L^2(\Omega )\), there exists a unique solution to the Poisson problem; find \(\rho \in \Psi \) such that
and there exists a positive constant \(c_R\) independent of \({\tilde{f}}\) and \(\rho \) such that
Definition 1
For \(\phi \in \Psi \), we define the Poisson projection \({\hat{\phi }}_h \in \Psi _h\) to \(\phi \) by
Theorem 3
(error estimates) Suppose that Hypothesis 1 holds true. For a given \(F\in H^1(0,T; \Psi ^\prime )\), let \(\{\phi (t) = \phi (\cdot ,t)\in \Psi ;~t\in (0,T) \}\) be the solution to problem (2). Suppose that Hypothesis 3 holds true. Let \(\Delta t\in {(0, 1)}\) be a time increment satisfying Hypothesis 2 and \(\{\phi _h^n\}_{n=1}^{N_T} \subset \Psi _h\) be the solution to scheme (9) with the initial condition \(\phi _h^0 = {\hat{\phi }}_h^0 \in \Psi _h\). Then, we have the following:
(i) There exist positive constants \(c_*\) and \(c_*^\prime \) independent of h and \(\Delta t\) such that
(ii) Suppose that additionally Hypothesis 4 holds. Then, there exists a positive constant \({\bar{c}}_*\) independent of h and \(\Delta t\) such that
4 Proofs
4.1 Proof of Theorem 1
We first note that due to Proposition 1-(i)
hold for any \(\rho \in \Psi \) and \(n=1,\dots , N_t\). We substitute \(1 \in \Psi _h\) into \(\psi _h\) in scheme (9) in the following.
We prove (i) by induction.
(I) Initial steps (\(n=0, 1\)): Since (10) with \(n=0\) is trivial, we prove it for \(n = 1\). We have
Hence, (10) holds for \(n=0, 1\).
(II) General steps: Let \(m\in \{2,\ldots , N_T\}\) and suppose that (10) holds true for \(n=m-1\). Then, we obtain (10) for \(n=m\) as follows:
From (I) and (II) the proof of (i) is completed.
We prove (ii) by induction.
(I’) Initial steps (\(n=0, 1\)): The property (11) is obvious for \(n=0, 1\), cf. (I) in the proof of (i).
(II’) General steps: Let \(m\in \{ 2, \ldots , N_T\}\) and assume that (11) holds true for \(n = m-1\) and \(m-2\), we prove that (11) also does for \(n=m\). From (9b) with \(f=0\), \(g=0\) and the induction assumption, we obtain (11) with \(n=m\) as follows:
From (I’) and (II’) the proof of (ii) is completed. \(\square \)
4.2 Proofs of Proposition 2 and Theorem 2
The proofs are given after stating two lemmas on a discrete Gronwall’s inequality and composite functions. The proof of the next lemma is given in Appendix A.1.
Lemma 1
Let \(a_i, i=0,1,2,\) be non-negative numbers with \(a_1 \ge a_2\), and \(\Delta t\in (0, 3/(4a_0)]\). Let \(\{x_n\}_{n\ge 0}\), \(\{y_n\}_{n\ge 1}\), \(\{z_n\}_{n\ge 2}\) and \(\{b_n\}_{n\ge 2}\) be non-negative sequences. Suppose that
holds. Then, it holds that
where \(a_*:=a_0+a_1+a_2\).
We recall some results concerning the evaluation of composite functions, which are mainly due to Lemma 4.5 in [1] and Lemma 1 in [15].
Lemma 2
[1, 15, 29, 34] Let a be a function in \(W^{1,\infty }_0(\Omega )^d\) satisfying \(\Delta t \Vert a\Vert _{1,\infty } \le 1/4\) and consider the mapping \(X_1(a,\Delta t)\) defined in (6). Then, the following inequalities hold.
Proof of Proposition 2
The equation (9b) can be written as
for \(n \ge 2\), where \(I_h^n \in \Psi _h^\prime \) with the norm \(\Vert \cdot \Vert _{\Psi _h} :=\Vert \cdot \Vert _{\Psi } \ (= \Vert \cdot \Vert _{H^1(\Omega )})\) is defined for \(n \in \{ 2, \ldots , N_T\}\) by
We prove (i). Substituting \(\phi _h^n \in \Psi _h\) into \(\psi _h\) in (23), we have
from the estimates, thanks to Young’s inequality and an identity in [33] for \(({\bar{D}}_{\Delta t}^{(2)}\phi _h^n, \phi _h^n)\),
The inequality (24) and Lemma 1 with
imply
which completes the proof of (i).
Next we prove (ii). Substituting \({\bar{D}}_{\Delta t}^{(2)}\phi _h^n \in \Psi _h\) into \(\psi _h\) in (23), we have
from the estimates
From the inequality (28), applying Lemma 1 with
and using the result of (i), we obtain
which completes the proof of (ii). \(\square \)
Proof of Theorem 2
We employ Proposition 2 for the proof. For the first step, \(n=1\), scheme (9a) can be written as
where \(I_h^1 \in \Psi _h^\prime \) is defined by
We first prove (i). Substituting \(\phi _h^1 \in \Psi _h\) into \(\psi _h\) in (30), and noting that
we have
which implies
The result (14) is obtained by combining (31) with Proposition 2-(i).
We next prove (ii). Substituting \({\bar{D}}_{\Delta t}^{(1)}\phi _h^n \in \Psi _h\) into \(\psi _h\) in (30), and noting that
we have
which implies
and, by taking into account (31) with \(g=0\),
The result (15) is obtained by combining (32) with Proposition 2-(ii). \(\square \)
4.3 Proof of Theorem 3
Error estimates for the Poisson projection are summarized in the following lemma.
Lemma 3
[12] Let \(\Psi _h\) be the finite element space defined in (5) with polynomial degree \(k\in {\mathbb {N}}\). Then, we have the following.
(i) There exists a positive constant c independent of h such that
(ii) Under Hypothesis 4, there exists a positive constant \(c^\prime \) independent of h such that
The next lemma shows the truncation error of second order in time for the time-discretization of \(\partial \phi /\partial t + \nabla \cdot ( u\phi )\), and plays an important role in the proof of Theorem 3.
Lemma 4
(truncation error) Suppose that Hypothesis 1 holds true. Assume \(\phi \in Z^3\). Suppose that Hypothesis 2 holds true. Then, there exists a positive constant \(c=c_1\) independent of \(\Delta t\) such that
Proof
Let \(n\in \{2, \ldots , N_T\}\) be fixed arbitrarily. From a simple calculation, the two Jacobians, \(\gamma ^n\) and \({\tilde{\gamma }}^n\), are written as
where \(\delta _i: \Omega \times (0,T) \rightarrow {\mathbb {R}}\), \(i = 1, 2\), are defined by
with the estimates \(\Vert \delta _i \Vert _{C(L^\infty )} \le c_1\), \(i=1, 2\). The relations (34) imply the key identity
Let us introduce the notations
Applying the identities
for \({\rho } (s) = \phi (y(\cdot ,s),t(s))\) we have the next expressions of \(O(\Delta t^2)\),
We evaluate \(\Vert I_i^n\Vert _{L^2(\Omega )}\), \(i=1,\ldots ,4\), as follows:
where for the last inequality in the estimate of \(\Vert I_4^n\Vert \), we have employed the inequality,
From the identity (35) and estimates (36), we obtain
which completes the proof. \(\square \)
Remark 6
([35]) For any \(n\in \{1, \ldots , N_T\}\), there exists a positive constant \(c = c_1\) independent of \(\Delta t\) such that
Remark 7
Lemma 4 and Remark 6 with \(u = 0\) imply that
Before the proof of Theorem 3, we prepare notations, equations and two lemmas to be employed. Let \(\{\phi (t) = \phi (\cdot ,t) \in \Psi ;\ t\in [0,T]\}\) be the solution to problem (2), and for each \(t\in [0, T]\), let \({\hat{\phi }}_h(t) = {\hat{\phi }}_h (\cdot , t) \in \Psi _h\) be the Poisson projection to \(\phi (t)\), cf. Definition 1. Let \(\{\phi _h^n\}_{n=1}^{N_T} \subset \Psi _h\) be the solution to scheme (9) with \(\phi _h^0 = {\hat{\phi }}_h^0 \in \Psi _h\). We introduce the two functions \(e_h^n\) and \(\eta (t)\) defined by
for \(n\in \{0, \ldots , N_T\}\) and \(t\in [0, T]\). Then, the series \(\{e_h^n\}_{n=0}^{N_T} \subset \Psi _h\) satisfies
for \(n \in \{ 1, \ldots , N_T \}\), where \(R_h^n \in \Psi _h^\prime \) is defined by
We summarize some estimates to be used in the proof of Theorem 3 in the next two lemmas. Their proofs are given in Appendix A.2 and A.3. The first lemma provides estimates for \(R_h^n\) and \(\eta ^n\) and the second lemma provides estimates for \(e_h^1\).
Lemma 5
Suppose that Hypotheses 1, 2 and 3 hold true. Assume \(\Delta t \in {(0, 1)}\). Then, we have the following.
(i) It holds that
(ii) Under Hypothesis 4, the estimates of \(\Vert \eta (\cdot ,t)\Vert \), \(\Vert R_{h2}^n\Vert _{\Psi _h^\prime }\) and \(\Vert R_{h3}^n\Vert _{\Psi _h^\prime }\) are given as
Remark 8
Hypotheses 1 and 2 are not needed for the estimates of (39a), (39b), (39e), (40a) and (40c).
Lemma 6
Suppose that Hypotheses 1, 2 and 3 hold true. Then, we have the following.
Now, we give the proof of the error estimates.
Proof of Theorem 3
Considering the equation (38) for \(e_h\), applying Proposition 2-(i) and (ii), and taking into account the fact \(e_h^0 = 0\), we have
We prove (i). From Lemma 5-(i), it holds that:
Combining (41a) and (44) with (42), we obtain
which implies the error estimate (17a) of (i), as
For the error estimate (17b), we have
Noting the estimate
we obtain the estimate (17b) of (i), as
We next prove (ii). Under Hypothesis 4, we have, from Lemma 5-(ii),
Combining (41a) and (48) with (42) and taking into account Lemma 3-(ii), we obtain
which completes the proof of (ii). \(\square \)
5 Numerical Results
In this section we verify the theoretical orders of convergence from Theorem 3 in numerical experiments. To this end we solved an example problem by scheme (9) in a finite element space of polynomial order \(k=1\). As initial data we set \(\phi _h^0 = \Pi _h\phi ^0\) using the Lagrange interpolation operator \(\Pi _h:C({\bar{\Omega }}) \rightarrow \Psi _h\), and note that this choice of \(\phi _h^0\) does not cause any loss of convergence order in Theorem 3. For the computation of the integrals appearing in the scheme we employed numerical quadrature formulae of degree nine for \(d=1\) (five points) and degree five for \(d=2\) (seven points) and \(d=3\) (fifteen points) [36]. While higher order quadrature formulae can improve numerical results of Lagrange-Galerkin methods, cf., e.g., [6, 14], we do not consider them in this paper. The linear systems were solved using the conjugate gradient method and meshes were generated using FreeFem++ [19].
Example 1
In problem (1), for \(d=1,2,3\), we set \(\Omega = (-1,1)^d\), \(T=0.5\), \(f=0\), \(g=0\), and
where \(\{e_i\}_{i=1}^d \subset {\mathbb {R}}^d\) is the standard basis in \({\mathbb {R}}^d\). The function \(\phi ^0\) is given according to the exact solution
The viscosity constant is set \(\nu = 10^{-2}\) if not otherwise noted.
We applied scheme (9) to Example 1 and computed the errors
for \(Y=\ell ^\infty (L^2)\), \(\ell ^2(H^1_0)\), \(\ell ^{\infty }(H^1_0)\), where \(\Vert \phi \Vert _{\ell ^2(H^1_0)} :=\Vert \nabla \phi \Vert _{\ell ^2(L^2)}\), \(\Vert \phi \Vert _{\ell ^\infty (H^1_0)} :=\Vert \nabla \phi \Vert _{\ell ^\infty (L^2)}\) and \(\Pi _h:C({\bar{\Omega }}) \rightarrow \Psi _h\) is the Lagrange interpolation operator. Tables 1–11 show the errors and the corresponding experimental orders of convergence (EOCs)Footnote 1 after grid refinement. The number N in the tables denotes the division number of the domain in each space dimension determining the mesh, whose size is taken as \(h :=2/N\). We coupled time increment and mesh size by \(\Delta t = c h^p\) and varied the constant c and the exponent p in the tables to see the theoretical convergence orders. According to Theorem 3 we expected to see experimental convergence orders 2 (\(E_{\ell ^\infty (L^2)}\)), 1 (\(E_{\ell ^2(H^1_0)}\)) and 1 (\(E_{\ell ^\infty (H^1_0)}\)) for \(p=1\), 2 (\(E_{\ell ^\infty (L^2)}\)), 2 (\(E_{\ell ^2(H^1_0)}\)) and 3/2 (\(E_{\ell ^\infty (H^1_0)}\)) for \(p=1/2\) and 2 (\(E_{\ell ^\infty (L^2)}\)), 3/2 (\(E_{\ell ^2(H^1_0)}\)) and 3/2 (\(E_{\ell ^\infty (H^1_0)}\)) for \(p=2/3\). The EOCs in the tables either agree with or exceed our expectations and therefore support our theoretical results. To see \(\Delta t\)-convergence for a fixed \(h~(=2/256)\) and h-convergence for a fixed \(\Delta t~(=0.01)\), we present Tables 12 and 13, respectively, which further support the convergence rates in Theorem 3.
The tables, i.e., Tables 1–13, moreover show a low relative loss of mass,
which decreases as the mesh is refined. Furthermore we computed the error formulas
for \(\Delta t = 4h\) shown in Table 14 providing additional information on the error of mass within the computation and throughout all time steps. Both \(E_{\text {mass}}^\prime \) and \(E_{\text {mass}}^{\prime \prime }\) also decrease as the mesh is refined.
These results indicate that mass is lost only due to numerical integration and Lagrange interpolation of the exact solution and thus support the mass-preserving property of the scheme (Theorem 1). When the viscosity \(\nu \) is decreased to \(\nu = 10^{-3}\) or \(10^{-4}\), we observe a reduction in the EOC in \(\ell ^\infty (L^2)\) to orders smaller than 2 for some N but still larger than 1, and almost no effect in the EOCs in \(\ell ^2(H^1_0)\) and \(\ell ^\infty (H^1_0)\), as we show in Tables 15 and 16.
We further present numerical solutions for \(d=2\) and 3 in Fig. 1.
6 Conclusions
We have presented a mass-preserving two-step Lagrange–Galerkin scheme of second order in time for convection-diffusion problems. Its mass-preserving property is achieved by the Jacobian multiplication technique, and its accuracy of second order in time is obtained based on the idea of the multistep Galerkin method along characteristics. For the first time step, we have proposed to employ a mass-preserving scheme of first order in time. This construction is efficient and does not decrease the convergence orders in the \(\ell ^\infty (L^2)\)- and \(\ell ^2(H^1_0)\)-norms.
Both main advantages of Lagrange–Galerkin methods, the CFL-free robustness for convection-dominated problems and the symmetric and positive coefficient matrix of the resulting system of linear equations, are kept in our scheme. Additionally, our scheme has a mass-preserving property as proved in Theorem 1. We have proved unconditional stability without any stabilization parameter in Theorem 2, and error estimates of second order in time in Theorem 3. For the error estimates two key lemmas on the truncation error analysis of the material derivative in conservative form, cf. Lemma 4, and a discrete Gronwall inequality for multistep methods, cf. Lemma 1, have been prepared.
We summarize the shown convergence orders as follows. The order in the \(\ell ^\infty (L^2)\cap \ell ^2(H^1_0)\)-norm is \(O(\Delta t^2 + h^k)\), and the order in the \(\ell ^\infty (L^2)\)-norm is \(O(\Delta t^2 + h^{k+1})\) if the duality argument can be employed. We have also proved the convergence order \(O(\Delta t^{3/2}+h^k)\) in the discrete \(\ell ^\infty (H^1_0)\)- and \(H^1(L^2)\)-norm, which will be useful when we apply the scheme to, e.g., the Navier–Stokes equations. We have presented numerical results in one-, two- and three-dimensions, which have supported the theoretical convergence orders.
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Notes
We used the formula \(\text {EOC}=\log (E_2/E_1) / \log (\Delta t_2/\Delta t_1)\) for errors \(E_1\), \(E_2\) and time increments \(\Delta t_1\), \(\Delta t_2\) from two consecutive table rows.
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Open Access funding enabled and organized by Projekt DEAL. This work was supported by JSPS KAKENHI Grant Numbers JP18H01135, JP19F19701, JP20H01823, JP20KK0058, and JP21H04431, JST CREST Grant Number JPMJCR2014, and JST PRESTO Grant Number JPMJPR16EA. NK was supported by the JSPS Postdoctoral Fellowships for Research in Japan (Standard).
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Appendix
Appendix
1.1 Proof of Lemma 1
From the assumption (20), there exists a non-negative sequence \(\{{\tilde{z}}_n\}_{n\ge 2}\) such that
where \({\tilde{z}}_n\) satisfies
Let p and q, \(p < q\), be the roots of quadratic equation \(f(x) :=(3/2-a_0\Delta t) x^2 - (2+a_1\Delta t) x + (1/2 -a_2\Delta t) = 0\), and let \(\lambda :=2/3\) and \(D :=(2+a_1\Delta t)^2 - (3-2a_0\Delta t)(1-2a_2\Delta t)\) \((= 1+2(a_0+2a_1+3a_2)\Delta t + (a_1^2-4a_0a_2)\Delta t^2)\). The numbers p and q have the properties
which are obtained from \(f(1) = - (a_0+a_1+a_2)\Delta t \le 0\), \(f(-1) = 4 + (- a_0 + a_1 - a_2) \Delta t \ge {13/4} > 0\), \(a_0\Delta t \le 3/4\), \(1 \le D \le [1+(a_0+2a_1+3a_2)\Delta t]^2\), and
Let \(n\ge 2\) be fixed arbitrarily. Then, we have
which imply
Multiplying (A.2a) by q and (A.2b) by p and subtracting the second equation from the first, we get
It is noted here that
where the following inequality has been employed:
This inequality holds obviously from the first property in (A.1) for \(p\ge 0\) or for \(p < 0\) and an even number k. For \(p < 0\) and an odd number k, it is proved by induction, and the key inequality in the induction is
Combining (A.1) and (A.4) with (A.3) and noting that \(0 \le -pq/(q-p) \le q/(q-p) \le 1\) for \(p \in (-1,0)\) and \(-pq/(q-p) \le 0 < 1\) for \(p \in [0,1)\), we obtain
which completes the proof.
1.2 Proof of Lemma 5
We prove (i). For the estimate (39a), from the next calculations,
and Lemma 3-(i), we obtain the inequalities as
For the estimate (39b), noting that
Thus, we obtain (39b). From Lemma 4 and Remark 6, the estimate of (39c) follows. Since we have
the estimate (39d) is obtained as
The estimate (39e) is obvious from (39a). Using a similar evaluation to (39d) with some modifications, we get (39f) by
We prove (ii). The estimate (40a) is obvious from Lemma 3-(ii). We evaluate \(\Vert R_{h2}^n\Vert _{\Psi _h^\prime }\). Recalling the calculation of \(\Vert R_{h2}^n\Vert _{\Psi _h^\prime }\), cf. (A.7) and (A.8), in the proof of (i), and noting that
for \(n \ge 1\), we have the following estimates,
which complete the proof of (40b). The estimate of (40c) is obvious under Hypothesis 4 from Lemma 3-(ii).
1.3 Proof of Lemma 6
We prove (i). From Lemma 5-(i), it holds that
The equation (38) with \(n=1\) is rewritten as
from \(e_h^0 = 0\) and, therefore, \(\frac{1}{\Delta t} (e_h^1 - e_h^0 \circ X_1^n\gamma ^n) = {\bar{D}}_{\Delta t}^{(1)}e_h^1\). Substituting \(e_h^1\) into \(\psi _h\) in (A.11), dropping the positive term \(a_0( e_h^1, e_h^1 )\), and using \(e_h^0 = 0\) and \(\langle R_h^1, e_h^1 \rangle \le \Vert R_h^1\Vert \Vert e_h^1\Vert \), we have
where for the last inequality we have employed
Again, substituting \(e_h^1\) into \(\psi _h\) in (A.11), and using \(e_h^0 = 0\) and \(\langle R_h^1, e_h^1 \rangle \le \Vert R_h^1\Vert \Vert e_h^1\Vert \), we have
which implies (41a).
Substituting \({\bar{D}}_{\Delta t}^{(1)}e_h^1\) into \(\psi _h\) in (A.11) and using the estimates,
we get
which implies (41b).
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Futai, K., Kolbe, N., Notsu, H. et al. A Mass-Preserving Two-Step Lagrange–Galerkin Scheme for Convection-Diffusion Problems. J Sci Comput 92, 37 (2022). https://doi.org/10.1007/s10915-022-01885-w
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DOI: https://doi.org/10.1007/s10915-022-01885-w