1 Introduction

Apart from their application in fluid flow problems convection–diffusion equations have recently been extensively used in the modeling of chemical and biological processes such as pollutant transport, immune system dynamics and cancer growth, see e.g. [2, 28]. In many of these applications, migration of cells or transport of solutes in fluids are involved, which leads to convection-dominant problems. In these cases the Galerkin finite element scheme can produce oscillating solutions. Hence, a plethora of extended and alternative numerical methods have been developed to perform stable computation, e.g., upwind methods [3, 23, 34] and characteristics methods [17, 18, 30, 35]. Among the latter ones the Lagrange–Galerkin (LG) method has been shown to be an effective and efficient method to deal with convection-dominated problems, see for example [4,5,6, 19, 20, 31, 32]. Not only the convection-dominant nature but also the rich dynamics of the biological applications, which include traveling waves and aggregation phenomena, pose a challenge for the numerical schemes. Different approaches related to mesh adaptation have been recently proposed to improve the accuracy in these cases, including a mass-transport approach for the one-dimensional problem [12] and adaptive mesh refinement [1, 25].

The application of the LG method to convection–diffusion problems offers the advantage that the time step size is not constrained by traditional CFL-type conditions. However, it is important to note that while a time step restriction is indeed necessary, this restriction is not directly proportional to the spatial discretization, as is typically the case with CFL conditions. This flexibility has motivated the use of the LG method in solving the Navier–Stokes equations and models of viscoelastic fluid flow, to name only a few [26, 27, 29]. Various LG methods have been proposed for convection–diffusion problems: Some have been concerned with higher order approximations in time using single- and multi-step methods, e.g., [4, 5, 7]. Others have focused on maintaining the mass balance on the discrete level, e.g. [16, 32]. In [20, 32] such mass-preserving LG schemes of first- and second-order in time have been proposed and error estimates have been provided. As the LG scheme relies on an upwind-interpolation of the numerical solution that follows the velocity field backwards in time a promising approach is to introduce mesh movement along the velocity field. From a computational point of view this might ease the identification of the upwind points and reduce the interpolation error.

Different kinds of moving mesh methods have been considered to numerically solve convection–diffusion problems. A common approach is the redistribution of mesh cells according to a monitor function that depends on local features of the numerical solution or an a posteriori error estimator, see e.g. [1]. In other approaches separate moving mesh PDEs and transformations obtained from the solution of the Monge–Kantorovich problem are used, see [22, 33]. In the context of hyperbolic balance laws and fluid dynamics a variety of schemes, which also entail mesh movement, has been derived from the Lagrangian formulation of the problem, such as the hydrodynamic GLACE scheme [11] and the arbitrary Lagrangian–Eulerian finite element method [13, 36]. In this context high order of accuracy has been achieved by adopting high order essentially non-oscillatory reconstructions, see e.g., [9, 14]. In some applications error estimates have been derived taking the movement of the mesh into account, e.g. [15, 21]. We consider here a Lagrangian grid approach based on a dynamical system for the nodal points that both exhibits significant benefits over static grids in numerical simulations and allows for an error analysis of the extended LG scheme.

In this work we are concerned with a moving mesh approach within LG schemes of first- and second-order in time and corresponding error estimates. We introduce the Lagrange–Galerkin Moving Mesh (LGMM) schemes, which combine the LG schemes derived in [20, 32] with a moving mesh, in order to improve the performance and efficiency over the original LG schemes with fixed mesh. The mesh movement we consider is inspired by [12]; although we focus on the one-dimensional case in this work our mesh movement is expressed in a form suitable for higher dimensional cases. We derive a condition under which the mesh movement is applicable. We generalize the mass-conservation property and stability results from the static grid versions of the schemes to the new LGMM schemes. Based on the idea of temporal derivatives on deforming grids in [24] we derive bounds for the time derivative of the dynamic interpolation operator, which then allow us to prove optimal error estimates for the LGMM scheme on piecewise linear elements in the \(\ell ^\infty (L^2) \cap \ell ^2(H_0^1)\) norm. Moreover, we present numerical experiments that verify the error estimates. They further show that in case of aggregation the LGMM method eliminates oscillations of the numerical solution that the LG method produces.

The rest of the paper is organized as follows: In Sect. 2 we present the mass-preserving LG schemes of first- and second-order in time for the convection–diffusion problem. This scheme is equipped with a mesh moving technique in Sect. 3, for which we state various properties. In Sects. 4 and 5, we provide the main results concerned with the mass-conservation property, the stability, and the error estimates for the schemes of order one and two, respectively, which are afterwards proven in Sect. 6. To show the advantages of the LGMM schemes, two numerical simulations are given in Sect. 7, followed by the conclusions in Sect. 8.

2 Lagrange–Galerkin Schemes

2.1 Statement of the Problem

Let \(\Omega =(a,b)\) be a bounded interval in \(\mathbb {R}\). We denote by \(\Gamma {:}{=}\partial \Omega \) the two point boundary of \(\Omega \) and by T a positive constant. In this paper we use the Lebesgue spaces \(L^2(\Omega ),L^{\infty }(\Omega )\) and the Sobolev spaces \(W^{m,p}(\Omega ),W^{1,\infty }_0(\Omega ),H^m(\Omega ),H^1_0(\Omega )\), for \(m\in \mathbb {N}\cup \{0\}\) and \(p\in [1,\infty ]\). We use the notation \((\cdot ,\cdot )\) to represent the \(L^2(\Omega )\) inner products for both scalar and vector-valued functions. The norm in \(L^2(\Omega )\) is simply denoted as \( \Vert \cdot \Vert {:}{=}\Vert \cdot \Vert _{L^2(\Omega )}\). For any normed space Y with norm \( \Vert \cdot \Vert _Y\), we define the function spaces \(H^m(0,T;Y)\) and \(C^0(0,T;Y)\) consisting of Y-valued functions in \(H^m(0,T)\) and \(C^0([0,T])\), respectively. For the two real numbers \(t_0<t_1\) we introduce the function space

$$\begin{aligned} Z^m(t_0,t_1){:}{=}\{\psi \in C^j(t_0,t_1;H^{m-j}(\Omega ));j=0,\dots ,m, \Vert \psi \Vert _{Z^m(t_0,t_1)}<\infty \}, \end{aligned}$$

with the norm

$$\begin{aligned} \Vert \cdot \Vert _{Z^m(t_0,t_1)}{:}{=}\left( \sum _{j=1}^m \Vert \cdot \Vert ^2_{C^j(t_0,t_1;H^{m-j}(\Omega ))}\right) ^{1/2}, \end{aligned}$$

and set \(Z^m{:}{=}Z^m(0,T)\). We often omit \(\Omega \) and [0, T] if there is no confusion and write, e.g., \(C^0(L^\infty )\) in place of \(C^0([0,T];L^\infty (\Omega ))\). Although we are concerned with a one-dimensional domain we use the general notations \(\nabla {:}{=}\partial _x\), \(\nabla \cdot {:}{=}\partial _x\), \(\Delta {:}{=}\partial _x^2\), and \(\frac{\partial }{\partial n} {:}{=}n \partial _x\) to refer to spatial derivatives in order to allow for a straightforward application of the multi-dimensional theory for LG schemes. We use c and C (with or without subscript or superscript) to denote generic positive constant independent of discretization parameters and solutions.

We consider a convection–diffusion problem, in which we aim to find \(\phi :\Omega \times (0,T) \rightarrow \mathbb {R}\) such that

$$\begin{aligned} \frac{\partial \phi }{\partial t}+\nabla \cdot (u\phi )-\nu \Delta \phi&=f \quad \text {in }\Omega \times (0,T), \end{aligned}$$
(1a)
$$\begin{aligned} \nu \frac{\partial \phi }{\partial n}-\phi u \cdot n&=g \quad \text {on }\Gamma \times (0,T),\end{aligned}$$
(1b)
$$\begin{aligned} \phi&=\phi ^0 \quad \text { in }\Omega ,\text {at }t=0, \end{aligned}$$
(1c)

where \(u:\Omega \times (0,T)\rightarrow \mathbb {R}\), \(f:\Omega \times (0,T)\rightarrow \mathbb {R}\), \(g:\Gamma \times (0,T)\rightarrow \mathbb {R}\) and \(\phi ^0:\Omega \rightarrow \mathbb {R}\) are given functions, \(n:\Gamma \rightarrow \{ \pm 1\}\) is the outward unit normal vector and \(\nu >0\) is the diffusion coefficient.

Let \(\Psi {:}{=}H^1(\Omega )\) and \(\Psi '\) be the dual space of \(\Psi \). The weak formulation corresponding to problem (1) is to find \(\{\phi (t)=\phi (\cdot ,t)\in \Psi ; t\in (0,T)\}\) such that for \(t\in (0,T)\) the variational equality

$$\begin{aligned} \left( \frac{\partial \phi }{\partial t}(t),\psi \right) +a_0(\phi (t),\psi )+a_1(\phi (t),\psi ;u(t))=\langle F(t),\psi \rangle , \quad \forall \psi \in \Psi \end{aligned}$$
(2)

holds in addition to \(\phi (0)=\phi ^0\). The bilinear forms \(a_0(\cdot ,\cdot )\) and \(a_u(\cdot ,\cdot )=a_1(\cdot ,\cdot \,;u)\) and for \(t \in (0,T)\) in (2) are defined such that by

$$\begin{aligned} a_1(\phi ,\psi ;u) {:}{=}-(\phi ,u \nabla \psi ) \end{aligned}$$

and \(F(t)\in \Psi '\) is a functional given by

$$\begin{aligned}&\langle F(t),\psi \rangle =(f(t),\psi )+[g(t),\psi ]_{\Gamma }, \\&\quad [g(t),\psi ]_{\Gamma } {:}{=}\int _{\Gamma }g(t)\psi \, ds {= g(a, t)\psi (a) + g(b, t)\psi (b)}, \end{aligned}$$

for \(f(t)=f(\cdot ,t)\in L^2(\Omega )\) and \(g(t)=g(\cdot ,t)\in L^2(\Gamma )\).

By substituting \(1\in \Psi \) into \(\psi \) in (2), and integrating over (0, t), we derive the mass balance identity

$$\begin{aligned} \int _{\Omega }\phi (x,t) \, dx = \int _{\Omega }\phi ^0(x) \, dx+\int _{0}^{t} \int _{\Omega }f(x,\tau ) dx \, d\tau +\int _{0}^{t}\int _{\Gamma }g(s,\tau ) ds \, d\tau \end{aligned}$$
(3)

that holds for all \(t\in (0,T)\). This property is desired to be maintained also at the discrete level, which is indeed achieved by the Lagrange–Galerkin schemes of first- and second-order in time proposed in [32] and [20], respectively.

2.2 The First-Order Lagrange–Galerkin Scheme

Let \(\Delta t > 0\) be a time step size, \(t^n {:}{=}n\Delta t\) for \(n\in \mathbb {N}\cup \{0\}\) an equidistant discretization of the time domain and \(N_T {:}{=}\lfloor {T/\Delta t}\rfloor \). For a function \(\rho \) defined in \(\Omega \times (0,T)\) and \(0\le t^n\le T\) the function \(\rho (\cdot ,t^n)\) in \(\Omega \) is denoted by \(\rho ^n\). Let \(\mathcal {T}_h^n{:}{=}\{K^n\},\) \(n\in \{0,\ldots ,N_T\}\) be a time-dependent partition of \(\bar{\Omega }\) with \(K^n\) representing an element in \(\mathcal {T}_h^n\). Let \(h{:}{=}\max _{n=0,\dots ,N_T}\max _{K^n\in \mathcal {T}_h^n} \text {diam} (K^n)\) denote the global mesh size. Let \(\Psi _h^n \subset \Psi \) be a time varying finite element space defined by

$$\begin{aligned} \Psi _h^n {:}{=}\left\{ \psi _h\in C^0(\bar{\Omega });\ \psi _{h|K^n} \in P_1(K^n),\forall K^n\in \mathcal {T}_h^n\right\} , \end{aligned}$$

where \(P_1(K^n)\) is the space of linear polynomial functions on \(K^n\in \mathcal {T}_h^n\). Details about the spaces \(\Psi _h^n\) and \(\mathcal {T}_h^n\) are discussed in Sect. 3.1, the rest of this section as well as Sect. 2.3 address the first- and second-order schemes assuming \(\Psi _h^n\) is known.

For a given velocity \(v:\Omega \rightarrow \mathbb {R}\), we define the upwind point of x with respect to v and \(\Delta t\) using the mapping \(X_1(v,\Delta t):\Omega \rightarrow \mathbb {R}\),

$$\begin{aligned} X_1(v,\Delta t)(x) {:}{=}x-v(x)\Delta t. \end{aligned}$$

With respect to the velocity u we define the mapping \(X_1^n: \Omega \rightarrow \mathbb {R}\) and its corresponding Jacobian \(\gamma ^n: \Omega \rightarrow \mathbb {R}\) by

$$\begin{aligned} X_1^n(x) {:}{=}X_1(u^n,\Delta t)(x)=x-u^n(x)\Delta t, \qquad \gamma ^n(x) {:}{=}\det \left( \frac{\partial X_1^n}{\partial x}(x)\right) . \end{aligned}$$

Suppose an approximate function \(\phi _h^0\in \Psi ^0_h\) of \(\phi ^0\) is given. In the first-order Lagrangian moving mesh scheme we look for \(\{\phi _h^n\in \Psi _h^n;\ n=1,\dots ,N_T\}\) such that for \(n = 1,\ldots , N_T\) it holds

$$\begin{aligned} \left( \frac{\phi ^n_h-\phi ^{n-1}_h\circ X^n_1 \gamma ^n}{\Delta t}, \psi _h \right) + a_0( \phi _h^n, \psi _h ) = \langle F^n, \psi _h \rangle , \quad \forall \psi _h \in \Psi _h^n. \end{aligned}$$
(4)

The functional \(F^n\in (\Psi _h^n)^\prime \) on the right hand side of (4) is defined by

$$\begin{aligned} \langle F^n, \psi _h \rangle {:}{=}( f^n, \psi _h ) + [g^n, \psi _h]_\Gamma . \end{aligned}$$

2.3 The Second-Order Lagrange–Galerkin Scheme

To obtain a higher order discretization in time we define the additional mapping \(\tilde{X}_1^n:\Omega \rightarrow \mathbb {R}\) and its Jacobian \(\tilde{\gamma }^n:\Omega \rightarrow \mathbb {R}\) by

$$\begin{aligned} \tilde{X}_1^n(x) {:}{=}X_1(u^n,2\Delta t)(x)=x-2u^n(x)\Delta t, \qquad \tilde{\gamma }^n(x) {:}{=}\det \left( \frac{\partial \tilde{X}_1^n}{\partial x}(x)\right) . \end{aligned}$$

Suppose an approximation \(\phi _h^0\in \Psi ^0_h\) of \(\phi ^0\) is given. Then the second-order Lagrangian moving mesh scheme aims to find \(\{\phi _h^n\in \Psi _h^n;\ n=1,\dots ,N_T\}\) satisfying

$$\begin{aligned}&\left( \frac{\phi ^n_h-\phi ^{n-1}_h\circ X^n_1 \gamma ^n}{\Delta t}, \psi _h \right) + a_0( \phi _h^n, \psi _h ) = \langle F^n, \psi _h \rangle , \quad \forall \psi _h \in \Psi _h^n,\quad n=1, \end{aligned}$$
(5a)
$$\begin{aligned}&\left( \frac{3\phi ^n_h-4\phi ^{n-1}_h\circ X^n_1 \gamma ^n+ \phi ^{n-2}_h\circ \tilde{X^n_1}\tilde{\gamma ^n}}{2 \Delta t}, \psi _h \right) + a_0( \phi _h^n, \psi _h ) = \langle F^n, \psi _h \rangle , \nonumber \\&\quad \forall \psi _h \in \Psi _h^n, \quad n\ge 2. \end{aligned}$$
(5b)

In the following, we rewrite scheme (5) as

$$\begin{aligned} ( \mathcal {A}_{\Delta t} \phi _h^n, \psi _h ) + a_0( \phi _h^n, \psi _h ) = \langle F^n, \psi _h \rangle , \ \ \forall \psi _h \in \Psi _h^n, \end{aligned}$$
(6)

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 given by

$$\begin{aligned} \mathcal {A}_{\Delta t} \rho ^n&{:}{=}\left\{ \begin{aligned}&\mathcal {A}_{\Delta t}^{(1)} \rho ^n{:}{=}\frac{1}{\Delta t}\left( \rho ^n-\rho ^{n-1} \circ X^n_1 \gamma ^n\right) ,&n&= 1, \\&\mathcal {A}_{\Delta t}^{(2)} \rho ^n{:}{=}\frac{1}{2\Delta t} \left( 3\rho ^n-4\rho ^{n-1} \circ X_1^n \gamma ^n + \rho ^{n-2} \circ \tilde{X}_1^n \tilde{\gamma }^n \right) ,&n&\ge 2, \end{aligned} \right. \end{aligned}$$

We introduce some discrete norms in the following. Let Y be a normed space, \(m\in \{0,\dots ,N_T\}\) be an integer, and \(\{\rho ^n\}^{N_t}_{n=0}\subset Y\). We define the norms \( \Vert \cdot \Vert _{\ell ^\infty _m (Y)}\) and \( \Vert \cdot \Vert _{\ell ^2_m(Y)}\) by

$$\begin{aligned} \Vert \rho \Vert _{\ell ^\infty _m (Y)}{:}{=}\max _{n=m,\dots ,N_T} \Vert \rho ^n \Vert _{Y}, \quad \Vert \rho \Vert _{\ell ^2_m(Y)}{:}{=}\left( \Delta t \sum _{n=m}^{N_T} \Vert \rho ^n \Vert ^2_{Y}\right) ^{1/2}. \end{aligned}$$

Additionally, we define the following norm over the time varying finite element spaces

$$\begin{aligned} \Vert \rho \Vert _{\ell _m^2(\Psi _h^\prime )} {:}{=}\left( \Delta t \sum _{n=m}^{N_T} \Vert \rho ^n \Vert ^2_{(\Psi _h^n)^\prime }\right) ^{1/2}. \end{aligned}$$

If there is no confusion we omit the subscript i.e, \( \Vert \rho \Vert _{\ell ^\infty _1(Y)} {=}{:}\Vert \rho \Vert _{\ell ^\infty (Y)}\) and \( \Vert \rho \Vert _{\ell ^2_1(Y)} {=}{:}\Vert \rho \Vert _{\ell ^2(Y)}\).

3 Moving Mesh Method for the Lagrange–Galerkin Scheme

3.1 Moving Mesh

In this section the construction and evolution of the partitions \(\mathcal {T}_h^n\) is considered. To this end a moving mesh is employed that in this work is defined as follows.

Definition 1

For a given partition \(\{t^n:~n=0,\dots ,N_T\}\) of the time domain [0, T] a moving mesh of \(\Omega \times [0, T]\) is a set of points \(\{P_i^n:~i=1,\dots ,N_p,~n=0,\dots ,N_T\} \subset \bar{\Omega }\) that satisfy the monotonicity condition

$$\begin{aligned} a=P^n_1< P^n_2< \cdots < P^n_{N_p}=b, \qquad n \in \{0,\dots ,N_T\}. \end{aligned}$$
(7)

We refer to \(N_p \in \mathbb {N}\) as the number of moving mesh points.

A moving mesh allows us to define the partitions introduced in Sect. 2.2 more precisely as

$$\begin{aligned} \mathcal {T}_h^n = \{K_{i}^n:~i=1,\dots , N_p - 1 \}, \qquad K_{i}^n {:}{=}[P^n_i, P^n_{i+1}], \qquad n \in \{0,\dots ,N_T\} \end{aligned}$$

and therefore determines the nodal points of the finite element spaces \(\Psi _h^n\) for \(n\in \{0, \dots , N_T\}\). We note that for any fixed \(i\in \{1, \dots , N_p\}\) the series \(\{P_i^n\}_{n=0}^{N_T}\) can be considered a time discrete trajectory of the moving point \(P_i\). For our analysis we define the velocities of the moving mesh points as

$$\begin{aligned} w^n_i{:}{=}\frac{P^n_i-P^{n-1}_i}{\Delta t}, \qquad i\in \{1,\dots ,N_p\},\quad n\in \{1,\dots ,N_t\}, \end{aligned}$$
(8)

which allow us to introduce the time continuous point trajectories

$$\begin{aligned} P_i(t) {:}{=}P^{n-1}_i+w^n_i(t-t^{n-1}), \qquad i\in \{1,\dots ,N_p\}, \quad t\in [t^{n-1},t^n]. \end{aligned}$$
(9)

In addition, for \(t\in [0,T]\) we define w(x, t) as an extension of \(w^n_i\), by

$$\begin{aligned} & w(x,t){:}{=}\frac{P_{i+1}(t)-x}{P_{i+1}(t)-P_i(t)}w^n_i+\frac{x-P_i(t)}{P_{i+1}(t)-P_i(t)}w^n_{i+1},\nonumber \\ & \quad x \in [P_i(t), P_{i+1}(t)], \quad t\in [t^{n-1},t^n]. \end{aligned}$$
(10)

Note that \(w \in C^0([0,T];W^{1,\infty }_0(\Omega ))\). Also the basis functions of the finite element spaces \(\Psi _h^n\) can be naturally extended using the trajectories (9): for \(i\in \{1,\dots , N_p\}\) let \(\psi _i(\cdot , t)\) denote the unique function on \(\Omega \) that is affine linear restricted to the intervals \([P_j(t), P_{j+1}(t)]\) for \(j\in \{1, \dots , N_p -1 \}\) and satisfies \(\psi _i(P_j(t), t) = \delta _{ij}\). Then clearly \(\{ \psi _i(\cdot , t^n):~i=1,\dots ,N_p \}\) is a basis of \(\Psi _h^n\).

3.2 Moving Mesh Method

In this section we propose a moving mesh method that is used to obtain a moving mesh in the sense of Definition 1 and therefore determines the finite element spaces \(\Psi _h^n\) as described in Sect. 3.1. Suppose that the points \(P_1^0\), ..., \(P_{N_P}^0\) are given and the monotonicity condition (7) is satisfied for \(n=0\). The method we propose determines the position of the points \(P_i^n\) iteratively by employing a time discretization of the dynamical system

$$\begin{aligned} \frac{d {\tilde{P}}_i}{dt}(t) = u\bigl ( {\tilde{P}}_i(t), t \bigr )&+ \nu _M \sum _{j=1}^{N_p -1}\nabla \psi _i(t)|_{[{\tilde{P}}_j(t), {\tilde{P}}_{j+1}(t)]} \end{aligned}$$
(11)

with initial data \({\tilde{P}}_i(0)=P_i^0\) for \(i\in \{1, \dots , N_p\}\). Here, \(\nabla \) refers to the gradient with respect to the spatial variables, and the parameter \(\nu _M\ge 0\) accounts for regularization of the moving mesh. The dynamical system (11) generalizes the mass transport approach from [8, 12]: If \(d=1\), \(f=0\) and \(\nu _M=\nu \) hold it provides a semi-discrete scheme for the convection–diffusion equation (1a) in terms of the inverse cumulative distribution function of the state \(\phi \). Applying it to the nodal points and assuming an exact solution of (1a) yields an equidistribution of the mass of the solution, i.e. \(\int _{{\tilde{P}}_i(t)}^{{\tilde{P}}_{i+1}(t)} \phi \, dx = Const\) for all \(i\in \{1, \dots , N_p-1\}\) and \(t\ge 0\). By employing (11) in our LG scheme we aim to follow the mass movement due to convection and diffusion with the moving mesh. Note that in this setting the approach can also be used if \(f\ne 0\), in which case a parameter \(\nu _M\ne \nu \) might yield more accurate results.

Applying a linearly implicit time discretization to the continuous problem (11) gives rise to our moving mesh method: find \(\{P_i^n:~i=1,\dots ,N_p,~n=0,\dots ,N_T\}\) such that for \(n=1,\dots ,N_T\) it holds

$$\begin{aligned} \frac{P^{n}_i-P^{n-1}_i}{\Delta t}&=u^{n-1}(P^{n-1}_i)+\nu _M \frac{P^{n}_{i+1}-2P^{n}_i+P^{n}_{i-1}}{(P^{n-1}_i-P^{n-1}_{i-1})(P^{n-1}_{i+1}-P^{n-1}_i)},\nonumber \\&\qquad i= 2,\dots ,N_p-1, \end{aligned}$$
(12a)
$$\begin{aligned} P^{n}_1&=a,\qquad P^{n}_{N_p}=b, \end{aligned}$$
(12b)
$$\begin{aligned} \{P_i^0:~i=1,\dots ,N_p\}&\subset \bar{\Omega } \text { given };\qquad a=P^0_1< P^0_2<...< P^0_{N_p}=b. \end{aligned}$$
(12c)

The discretization has been constructed making use of the fact that for \(d=1\) it holds

$$\begin{aligned} \sum _{j=1}^{N_p -1}\nabla \psi _i|_{[{\tilde{P}}_j, {\tilde{P}}_{j+1}]} = \frac{1}{{\tilde{P}}_i - {\tilde{P}}_{i-1}} - \frac{1}{{\tilde{P}}_{i+1} - {\tilde{P}}_{i}} = \frac{{\tilde{P}}_{i+1} - 2 {\tilde{P}}_{i} + {\tilde{P}}_{i-1}}{({\tilde{P}}_i - {\tilde{P}}_{i-1}) ({\tilde{P}}_{i+1} - {\tilde{P}}_{i})}. \end{aligned}$$

The method is inspired by [12] and can be extended to higher dimensions in a straightforward way. In the case \(\nu _M= 0\), the transition from \(P_i^{n-1}\) to \(P_i^n\) due to (12) and the transition from \(P_i^{n-1}\) to \(X_1(u^{n-1}, \Delta t)(P_i^{n-1})\) describe movements in opposite directions. In particular, if the velocity field u is smooth we have \( P_i^{n-1} \approx X_1^n(P_i^n)\). Hence, a reduction of the computational costs to identify \(X_1^n(P_i^n)\) as well as a decrease of the corresponding interpolation error in the scheme are expected. The main idea of the LGMM method is, to combine the LG schemes (4) and (6) with the moving mesh method (12).

Remark 1

While the moving mesh method (12) leads to a well defined set of nodal points \(\{P_i^n:~i=1,\dots ,N_p,~n=0,\dots ,N_T\}\) it is not clear whether they constitute a moving mesh in the sense of Definition 1 since the condition (7) might not be satisfied.

Remark 2

To obtain the nodal points \(P_1^n,\dots , P_{N_p}^n\) from \(P_1^{n-1},\dots , P_{N_p}^{n-1}\) according to (12a) and (12b) a sparse linear system is solved. In general, the coefficient matrix of this system is not symmetric.

In fact we show that the method (12) results in a moving mesh for a suitable choice of \(\Delta t\), see Sect. 1. The other theoretical results we show in this work assume a given moving mesh. While it is important that a positive distance between neighbor points is maintained in the moving mesh, it needs to be also verified that this distance does not become too large. In particular, with respect to the error estimates that we present in the following section we are interested in the situation that the global mesh size h tends to 0. This can be realized by employing an equidistant mesh of size \(h_0\) at the initial time that is iteratively decreased by increasing the number of moving points and ensuring that the distance between neighboring points does not exceed \(C h_0\) over all time instances for a fixed constant \(C>0\). In practice, positive \(\nu _M\) in scheme (12) has resulted in a control over the maximal point distance. Next, we state several results concerning the moving mesh method (12). We begin by formulating the following hypotheses.

Hypothesis 1

The function u satisfies \(u\in C^0([0,T];W^{1,\infty }_0(\Omega ))\).

Hypothesis 2

The nodal points of the finite element spaces \(\Psi _h^0, \dots \Psi _h^{N_T}\) are given by a moving mesh.

Hypothesis 3

The solution \(\phi \) of problem (1) satisfies \(\phi \in Z^3 \cap H^2 (0,T;H^2(\Omega )) \cap H^1 (0,T;H^3(\Omega ))\).

Theorem 1

(Non-overlapping condition for the moving mesh method) Suppose that Hypothesis 1 holds true. Let \(C_0\in [0,1)\) be fixed, the set of nodal points \(\{P^n_i:~i=1,\dots ,N_p,~n=1,\dots ,N_T\}\) be given by method (12), and

$$\begin{aligned} \Delta t |u|_{C^0(W^{1,\infty }(\Omega ))}\le C_0, \end{aligned}$$
(13)

then the set of nodal points describes a moving mesh, i.e., it holds that for any \(n\in \{0,\dots ,N_T\}\)

$$\begin{aligned} P^{n}_i< P^{n}_j; \quad i< j; \quad i,j\in \{1,\dots ,N_p\}. \end{aligned}$$
(14)

Proof

Refer to Sect. 6.1.1. \(\square \)

Remark 3

Suppose that the nodal points of the finite element spaces \(\Psi _h^0, \dots \Psi _h^{N_T}\) are governed by the moving mesh method (12) then Hypothesis 2 is implied by condition (13) due to Theorem 1.

Next, we state two results that are necessary in order to derive the error estimates for the LGMM schemes. For \(f\in C^0(\bar{\Omega }), t\in [0,T]\), and the time dependent P1-basis functions \(\psi _i(x, t)\) for \(i\in \{1,\dots , N_p\}\) we define the time dependent Lagrange interpolation of f by

$$\begin{aligned} \left[ \Pi _h (t) f\right] (x){:}{=}\sum _{i=1}^{N_p} f(P_i(t))\psi _i(x,t). \end{aligned}$$
(15)

We also denote the difference operator \(\bar{D}_{\Delta t} f {:}{=}\frac{f^n - f^{n-1}}{\Delta t}\).

Theorem 2

Let \(\{\phi (t)=\phi (\cdot ,t)\in \Psi ; t\in (0,T)\}\) be the solution of problem (1). Suppose that Hypothesis 2 and Hypothesis 3 hold true. Then assuming \(w \in C^0(W^{1,\infty }_0(\Omega ))\) the following results hold.

  1. i)

    There exists a positive constant \(C=C(\Vert w \Vert _{C^0(L^\infty )})\) independent of \(\Delta t\) and h such that

    $$\begin{aligned} \left\| \bar{D}_{\Delta t}(\Pi ^n_h\phi ^n) - \frac{1}{\Delta t} \int _{t^{n-1}}^{t^{n}} \Pi _h(t) \frac{\partial \phi }{\partial t} (\cdot , t) dt \right\| \le \frac{C h}{\sqrt{\Delta t}} \Vert \phi \Vert _{L^2(t^{n-1},t^n ; H^2(\Omega ))}. \end{aligned}$$
    (16a)
  2. ii)

    For a positive constant \(C^\prime = C^\prime (\Vert w \Vert _{C^0(W^{1,\infty })})\) independent of \(\Delta t\) and h it holds

    $$\begin{aligned} \left\| \bar{D}_{\Delta t}(\Pi ^n_h\phi ^n) - \frac{1}{\Delta t} \int _{t^{n-1}}^{t^{n}} \Pi _h(t) \frac{\partial \phi }{\partial t} (\cdot , t) dt \right\| _{\Psi ^\prime } \le C h^2 \Vert \phi \Vert _{H^1(H^3)}. \end{aligned}$$
    (16b)

Proof

Refer to Sect. 6.1.2. \(\square \)

Remark 4

In the case of a fixed mesh (velocity \(w=0\)), the bounds on the right hand side of (16b) and (16a) are zero.

Corollary 1

Let \(\{\phi (t)=\phi (\cdot ,t)\in \Psi ; t\in (0,T)\}\) be the solution of problem (1). Suppose that Hypothesis 2 and Hypothesis 3 hold true. Define \(\eta (t){:}{=}\phi (t)-\Pi _h (t) \phi (t)\). Then for \(w \in C^0(W^{1,\infty }_0(\Omega ))\) there exist positive constants \(C=C(\Vert w \Vert _{C^0(L^\infty )})\) and \(C^\prime =C^\prime (\Vert w \Vert _{C^0(W^{1,\infty })})\) independent of \(\Delta t\) and h such that the following bounds hold

$$\begin{aligned} \left\| \bar{D}_{\Delta t}\eta ^n \right\|&\le \frac{C h}{\sqrt{\Delta t}} \Vert \phi \Vert _{H^1(t^{n-1},t^n;H^2(\Omega ))}, \end{aligned}$$
(17a)
$$\begin{aligned} \left\| \bar{D}_{\Delta t}\eta ^n \right\| _{\Psi ^\prime }&\le C' h^2 \left( \frac{1}{\sqrt{\Delta t}} \Vert \phi \Vert _{H^1(t^{n-1},t^n;H^2(\Omega ))} + \Vert \phi \Vert _{H^1(H^3)} \right) . \end{aligned}$$
(17b)

Proof

Refer to Sect. 6.1.3. \(\square \)

4 Results For The First-Order LGMM Scheme

In this section we state results for the scheme introduced in Sect. 2.2. We start by stating the following hypothesis.

Hypothesis 4

The time step size \(\Delta t\) satisfies the condition \(\Delta t |u|_{C^0(W^{1,\infty })}\le 1/8\).

Remark 5

Hypothesis 4 is not a CFL condition since the mesh size h is not included in the inequality. The time step size \(\Delta t\) can be chosen independently of h.

Proposition 1

(Mass preserving property of the first-order LGMM Scheme) Suppose that Hypotheses 1, 2 and 4 hold true. Let \(\{\phi ^n_h\}_{n=1}^{N_T}\) be the solution of the numerical scheme (4) for a given initial datum \(\phi _h^0\). Then it holds for \(n=0,1,\dots ,N_T\) that

$$\begin{aligned} \int _{\Omega }\phi ^n_hdx = \int _{\Omega }\phi ^0_h dx + \Delta t \sum _{i=1}^{n}\left( \int _{\Omega }f^i dx + \int _{\Gamma } g^i ds \right) . \end{aligned}$$
(18)

Proposition 2

(Stability of the first-order LGMM scheme) Suppose that Hypotheses 1, 2 and 4 hold true. Let \(F\in H^1(0,T;\Psi ')\) be given. For the given function \(\phi ^0_h \in \Psi _h\), let \(\{\phi ^n_h\}^{N_T}_{n=1} \subset \Psi _h\) be the numerical solutions of scheme (4). Then there exists a constant \(C>0 \) independent of h and \(\Delta t\) such that

$$\begin{aligned} \Vert \phi _h \Vert _{\ell ^\infty (L^2)} + \sqrt{\nu } \Vert \nabla \phi _h\Vert _{\ell ^2(L^2)} \le C \left( \Vert \phi ^0_h \Vert + \Vert F \Vert _{\ell ^2(\Psi '_h)}\right) . \end{aligned}$$
(19)

Proofs of Proposition 1 and Proposition 2

The proof of the two propositions follows directly from Theorem 1 and Theorem 2 in [32], respectively. For convenience we provide the proofs in Appendix D.1 and Appendix E.1. \(\square \)

Remark 6

The mass-preserving and stability properties of the first-order Lagrange-Galerkin scheme with fixed mesh (Theorem 1 and Theorem 2 of [32]) are maintained in the first-order LGMM scheme.

Theorem 3

(Error estimates for the first-order LGMM scheme) Suppose that Hypotheses 1, 2, 4, and 3 hold true. Let \(F\in H^1(0,T;\Psi ')\) be given. Assuming the initial datum \(\phi ^0_h=\Pi ^0_h\phi ^0\in \Psi _h\) let \(\{\phi (t)=\phi (\cdot ,t)\in \Psi ; t\in (0,T)\}\) be the solution of problem (1) and \(\{\phi ^n_h\}_{n=1}^{N_T}\) the numerical solutions of scheme (4). Then there exists a constant \(C>0 \) independent of h and \(\Delta t\) such that

$$\begin{aligned} \begin{aligned} \Vert \phi _h - \phi \Vert _{\ell ^\infty (L^2)} + \sqrt{\nu } \Vert \nabla (\phi _h-\phi )\Vert _{\ell ^2(L^2)}&\le C (\Delta t + h^2) \Vert \phi \Vert _{Z^2 \cap H^1(H^2) \cap H^1(H^3)}. \end{aligned} \end{aligned}$$
(20)

Proof

Refer to Sect. 6.2. \(\square \)

Remark 7

Using the bound (17a) instead of (17b) in the proof of Theorem 3 a first order bound that requires lower regularity of \(\phi \) is obtained. Namely, under the assumptions of Theorem 3 there exists a constant \(C>0\) independent of h and \(\Delta t\) such that

$$\begin{aligned} \Vert \phi _h - \phi \Vert _{\ell ^\infty (L^2)} + \sqrt{\nu } \Vert \nabla (\phi _h-\phi )\Vert _{\ell ^2(L^2)} \le C (\Delta t + h) \Vert \phi \Vert _{Z^2 \cap H^1(H^2) \cap H^1(H^2)}. \end{aligned}$$

5 Results For The Second-Order LGMM Scheme

The results in this section concern the second-order scheme introduced in Sect. 2.3.

Proposition 3

(Mass preserving property of the second-order LGMM scheme) Suppose that Hypotheses 1, 2 and 4 hold true. Let \(\{\phi ^n_h\}_{n=1}^{N_T}\) be the solution of the numerical scheme (6) for a given initial datum \(\phi _h^0\). Then, we have the following.

(i) It holds for \(n=1,2,\dots ,N_T\) that

$$\begin{aligned} \int _{\Omega }\left( \frac{3}{2} \phi ^n_h - \frac{1}{2} \phi ^{n-1}_h \right) dx = \int _{\Omega } \frac{1}{2} \bigl ( \phi ^0_h+\phi ^1_h \bigr ) dx + \Delta t \sum _{i=1}^{n}\left( \int _{\Omega }f^i dx + \int _{\Gamma } g^i ds \right) . \end{aligned}$$
(21)

(ii) Assume \(f=g=0\) additionally. Then, it holds for \(n=1,2,\dots ,N_T\) that

$$\begin{aligned} \int _{\Omega } \phi ^n_h dx = \int _{\Omega } \phi ^0_h dx. \end{aligned}$$
(22)

Proposition 4

(Stability for the Second-Order LGMM Scheme) Suppose that Hypotheses 1, 2 and 4 hold true. Let \(F\in H^1(0,T;\Psi ')\) be given. For a given function \(\phi ^0_h \in \Psi _h\) let \(\{\phi ^n_h\}^{N_T}_{n=1} \subset \Psi _h\) be the numerical solutions of scheme (6). Then, there exists a constant \(C>0\) independent of h and \(\Delta t\) such that

$$\begin{aligned} \Vert \phi _h \Vert _{\ell ^\infty (L^2)} + \sqrt{\nu } \Vert \nabla \phi _h\Vert _{\ell ^2(L^2)} \le C \left( \Vert \phi ^0_h \Vert + \Vert F \Vert _{\ell ^2(\Psi '_h)}\right) . \end{aligned}$$
(23)

Proofs of Proposition 3 and Proposition 4

The proof of both propositions follows from Theorem 1 and Theorem 2 in [20], respectively. For convenience we provide the proofs in Appendices D.2 and E.2. \(\square \)

Remark 8

Also in case of the second-order Lagrange–Galerkin scheme the mass-preserving and stability properties of the fixed mesh method (Theorem 1 and Theorem 2 of [20]) are maintained in the LGMM scheme.

Theorem 4

(Error Estimates the Second-Order LGMM Scheme) Suppose that Hypotheses 1, 2, 4, and 3 hold true. Let \(F\in H^1(0,T;\Psi ')\) be given. For a given function \(\phi ^0_h=\Pi ^0_h\phi ^0\in \Psi _h\) let \(\{\phi (t)=\phi (\cdot ,t)\in \Psi ; t\in (0,T)\}\) be the solution of problem (1) and \(\{\phi ^n_h\}_{n=1}^{N_T}\) be the numerical solutions of scheme (6). Then, there exists a constant \(C>0 \) independent of h and \(\Delta t\) such that

$$\begin{aligned} \begin{aligned} \Vert \phi _h - \phi \Vert _{\ell ^\infty (L^2)} + \sqrt{\nu } \Vert \nabla (\phi _h-\phi )\Vert _{\ell ^2(L^2)}&\le C (\Delta t^2 + h^2) \Vert \phi \Vert _{Z^3 \cap H^2(H^2)\cap H^1(H^3)}. \end{aligned} \end{aligned}$$
(24)

Proof

Refer to Sect. 6.3. \(\square \)

Remark 9

In analogy to Remark 7 in the proof of Theorem 4 the bound (17a) can be used instead of (17b) to obtain a first order bound that requires lower regularity of \(\phi \). Namely, under the assumptions of Theorem 4 there exists a constant \(C>0\) independent of h and \(\Delta t\) such that

$$\begin{aligned} \Vert \phi _h - \phi \Vert _{\ell ^\infty (L^2)} + \sqrt{\nu } \Vert \nabla (\phi _h-\phi )\Vert _{\ell ^2(L^2)} \le C (\Delta t^2 + h) \Vert \phi \Vert _{Z^3 \cap H^2(H^2)} \end{aligned}$$

Remark 10

In case of a static mesh the error estimate (24) is consistent with [20, Theorem 3 (ii)] except for the dependence on \(\Vert \phi \Vert _{H^1(H^3)}\). Taking into account Remark 4 the exact literature result can easily be recovered. The same is true for the relation between the error estimate (20) and [32, Theorem 3].

6 Proofs

In this section we provide proofs for the results stated in Sects. 3, 4 and 5.

6.1 Proofs of the Results Regarding the Moving Mesh

6.1.1 Proof of Theorem 1

We show property (14) inductively. Hence, suppose \(P^{n-1}_i < P^{n-1}_j;\) \( i < j;\) \( i,j\in \{1,\dots ,N_p\}\), we show that (14) holds true. Let \(h^{n-1}_i{:}{=}P^{n-1}_{i+1}-P^{n-1}_i\) for \(i \in \{1,\dots ,N_p-1\}\). It is sufficient to show that \(h^{n}_i > 0\) for \(i \in \{1,\dots ,N_p-1\}\). Shifting the index i in scheme (12a), we have

$$\begin{aligned} \frac{P^{n}_{i+1}-P^{n-1}_{i+1}}{\Delta t}=u^{n-1}(P^{n-1}_{i+1})+\nu _{M}\frac{P^{n}_{i+2}-2P^{n}_{i+1}+P^{n}_{i}}{(P^{n-1}_{i+1}-P^{n-1}_{i})(P^{n-1}_{i+2}-P^{n-1}_{i+1})}, \quad i= 1,\dots ,N_p-2. \end{aligned}$$
(25)

By subtracting (12a) from (25) we obtain

$$\begin{aligned} & \frac{h^{n}_{i}-h^{n-1}_{i}}{\Delta t}=u^{n-1}(P^{n-1}_{i+1})-u^{n-1}(P^{n-1}_{i})+ \nu _M \left[ \frac{h^{n}_{i+1}-h^{n}_{i}}{h^{n-1}_{i}h^{n-1}_{i+1}}-\frac{h^{n}_{i}-h^{n}_{i-1}}{h^{n-1}_{i-1}h^{n-1}_{i}}\right] ,\\ & \quad i= 2,\dots ,N_p-2. \end{aligned}$$

Rearranging the terms it follows for \(i=2,\dots ,N_p-2\)

$$\begin{aligned}&\left[ \frac{1}{\Delta t}+\nu _M\left( \frac{1}{h^{n-1}_{i}h^{n-1}_{i+1}}+\frac{1}{h^{n-1}_{i-1}h^{n-1}_{i}}\right) \right] h^{n}_i -\nu _M\frac{1}{h^{n-1}_{i}h^{n-1}_{i+1}}h^{n}_{i+1}-\nu _M\frac{1}{h^{n-1}_{i-1}h^{n-1}_{i}}h^{n}_{i-1} \nonumber \\&\quad =u^{n-1}(P^{n-1}_{i+1})-u^{n-1}(P^{n-1}_{i})+\frac{h^{n-1}_i}{\Delta t}. \end{aligned}$$
(26)

Using (13) we derive a lower bound of the right hand side in (26) as follows:

$$\begin{aligned}&u^{n-1}(P^{n-1}_{i+1})-u^{n-1}(P^{n-1}_{i})+\frac{h^{n-1}_i}{\Delta t} \ge \frac{h^{n-1}_i}{\Delta t}-|u^{n-1}(P^{n-1}_{i+1})-u^{n-1}(P^{n-1}_{i})| \nonumber \\&\quad \ge \frac{h^{n-1}_i}{\Delta t}-h^{n-1}_i|u|_{C^0(W^{1,\infty }_0)}=\frac{h^{n-1}_i}{\Delta t}\left( 1-\Delta t|u|_{C^0(W^{1,\infty }_0)}\right) >0. \end{aligned}$$
(27)

Note that from (12a) for \(i=2\) we have

$$\begin{aligned} \frac{P^{n}_2-P^{n-1}_2}{\Delta t}=u^{n-1}(P^{n-1}_2)+\nu _M\frac{h^{n}_2-h^{n}_1}{h^{n-1}_1h^{n-1}_2}, \end{aligned}$$

and from (12b) follows \(P^{n-1}_1=a\), which implies \( \frac{P^{n}_1-P^{n-1}_1}{\Delta t}=0\). Therefore, it holds

$$\begin{aligned} \left( \frac{1}{\Delta t}+\nu \frac{1}{h^{n-1}_{1}h^{n-1}_{2}}\right) h^{n}_1-\nu _M\frac{1}{h^{n-1}_{1}h^{n-1}_{2}}h^{n}_{2} =u^{n-1}(P^{n-1}_{2})-u^{n-1}(P^{n-1}_{1})+\frac{h^{n-1}_1}{\Delta t}. \end{aligned}$$
(28)

Similarly, from (12a) for \(i=N_p-1\) we have

$$\begin{aligned} \frac{P^{n}_{N_p-1}-P^{n-1}_{N_p-1}}{\Delta t}=u^{n-1}(P^{n-1}_{N_p-1})+\nu _M\frac{h^{n}_{N_p-1}-h^{n}_{N_p-2}}{h^{n-1}_{N_p-2}h^{n-1}_{N_p-1}} \end{aligned}$$

and from (12b) we obtain \(P^{n-1}_{N_p}=b\), which implies \( \frac{P^{n}_{N_p}-P^{n-1}_{N_p}}{\Delta t}=0\). Therefore, we have

$$\begin{aligned} & \left( \frac{1}{\Delta t}+\nu \frac{1}{h^{n-1}_{N_p-2}h^{n-1}_{N_p-1}}\right) h^{n}_{N_p-1} -\nu _M\frac{1}{h^{n-1}_{N_p-2}h^{n-1}_{N_p-1}}h^{n}_{N_p-2}\nonumber \\ & \quad =u^{n-1}(P^{n-1}_{N_p})-u^{n-1}(P^{n-1}_{N_p-1})+\frac{h^{n-1}_{N_p-1}}{\Delta t}. \end{aligned}$$
(29)

Proceeding as in (27) positivity of the right hand sides in both (28) and (29) follows. Combining (28), (26) and (29) yields a linear system with unknown variables \(h^{n}_1\), ..., \(h^n_{N_p-1}\) and a strictly diagonally dominant coefficient matrix, which is thus an M-matrix. Since an M-matrix A has the property that \(Ax>0\) implies \(x>0\), the solution of the linear system is positive, i.e., \(h^{n}_{1}\), ...,\(h^n_{N_p-1} > 0\), hence (14) follows. \(\square \)

6.1.2 Proof of Theorem 2

First, we state the following lemma, which plays an important role in the proof of Theorem 2. The proof of the lemma is given in Appendix A.

Lemma 1

Let \(\{\phi (t)=\phi (\cdot ,t)\in \Psi ; t\in (0,T)\}\) be the solution of problem (1) and suppose that Hypothesis 2 holds true. For \(x\in \bar{\Omega }\) and \(t\in [t^{n-1},t^n]\) we define

$$\begin{aligned} I (x,t) {:}{=}\sum _{i=1}^{N_p}\phi (P_i(t),t)\left[ \frac{\partial }{\partial t}\psi _i(x,t)\right] , \end{aligned}$$

where \(P_i(t)\) for \(i\in \{1,\dots ,N_p\}\) are the nodal point positions defined in (9) and \(\psi _i(x,t)\) for \(i\in \{1,\dots ,N_p\}\) denote the time extended P1 basis functions. We assume \(x\in [P_k(t), P_{k+1}(t)]\) for a \(k\in {\{1,\dots ,N_p -1\}}\). Then I(x, t) can be expressed as

$$\begin{aligned} I(x,t)=-\frac{\phi (P_{k+1}(t),t)-\phi (P_k(t),t)}{P_{k+1}(t)-P_k(t)}\left[ w^n(P_{k+1}(t))\psi _{k+1}(x,t)+w^n(P_k(t))\psi _k(x,t))\right] . \end{aligned}$$
(30)

Proof of Theorem 2

We first define the interpolation operators

$$\begin{aligned} \left( \Pi ^\ell _h\phi ^\ell \right) (x)&{:}{=}\sum _{i=1}^{N_p} \phi ^\ell (P_i^\ell )\psi ^\ell _i (x), \qquad \ell \in \{ n-1, n\}. \end{aligned}$$

Then we rewrite their difference as

$$\begin{aligned}&\left( \Pi ^n_h\phi ^n-\Pi ^{n-1}_h\phi ^{n-1}\right) (x) \nonumber \\&\quad =\int _{t^{n-1}}^{t^n}\frac{d}{dt} (\Pi _h(t)\phi (\cdot ,t))(x) dt \nonumber \\&\quad =\sum _{i=1}^{N_p}\int _{t^{n-1}}^{t^n}\frac{\partial }{\partial t} [\phi (P_i(t),t)\psi _i(x,t)] dt \nonumber \\&\quad =\sum _{i=1}^{N_p}\int _{t^{n-1}}^{t^n}\left( \left[ \frac{\partial }{\partial t}\phi (P_i(t),t)\right] \psi _i(x,t)+\phi (P_i(t),t)\left[ \frac{\partial }{\partial t}\psi _i(x,t)\right] \right) dt\nonumber \\&\quad =\sum _{i=1}^{N_p}\int _{t^{n-1}}^{t^n}\left( \left[ \frac{\partial \phi }{\partial t}(P_i(t),t) + \frac{d P_i}{d t}(t)(\nabla \phi )(P_i(t),t)\right] \psi _i(x,t)\right. \nonumber \\&\qquad \left. +\phi (P_i(t),t)\left[ \frac{\partial }{\partial t}\psi _i(x,t)\right] \right) dt\nonumber \\&\quad =\sum _{i=1}^{N_p}\int _{t^{n-1}}^{t^n}\left( \left[ \frac{\partial \phi }{\partial t}(P_i(t),t) + w^n(P_i(t))(\nabla \phi )(P_i(t),t)\right] \psi _i(x,t)\right. \nonumber \\&\qquad \left. +\phi (P_i(t),t)\left[ \frac{\partial }{\partial t}\psi _i(x,t)\right] \right) dt\nonumber \\&\quad =\int _{t^{n-1}}^{t^n} \Pi _h(t) \left[ \frac{\partial \phi }{\partial t} (\cdot ,t) + w^n(\cdot ) \nabla \phi (\cdot ,t)\right] (x) dt + \int _{t^{n-1}}^{t^n} I(x,t) dt. \end{aligned}$$
(31)

The rest of the proof concerns the last integral in (31). Without loss of generality let \(t\in [t^{n-1},t^n]\) and \(x\in K_k(t) {:}{=}[P_k(t), P_{k+1}(t)]\). For brevity we introduce the notations:

$$\begin{aligned} w^n_{k}&{:}{=}w^n(P^{n-1}_{k}), \qquad w^n_{k+1}{:}{=}w^n(P^{n-1}_{k+1}), \qquad h_k = P_{k+1}(t) - P_k(t), \\ \phi _{k}&{:}{=}\phi (P_{k}(t),t), \qquad \phi _{k+1}{:}{=}\phi (P_{k+1}(t),t). \end{aligned}$$

We’re now in the position to show i). Due to the Taylor expansions

$$\begin{aligned} \phi _{k+1} - \phi _k&=\phi (P_{k}(t)+h_k(t),t) - \phi (P_{k}(t),t)\\&=h_k(t) ( \nabla \phi ) (P_k(t),t) + h_k^2(t) \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_k(t)+s_1 h_k(t), t) ds_1 ds_0, \\ \phi _{k+1} - \phi _k&=\phi (P_{k+1}(t),t) - \phi (P_{k+1}(t)-h_k(t),t)\\&=h_k(t) ( \nabla \phi ) (P_{k+1}(t),t) - h_k^2(t) \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_{k+1}(t)-s_1 h_k(t), t) ds_1 ds_0. \end{aligned}$$

we obtain the identity

$$\begin{aligned}&\frac{\phi _{k+1} - \phi _k}{P_{k+1}(t)-P_k(t)}\left[ w^n_{k}\psi _{k}(x,t)+w^n_{k+1}\psi _{k+1}(x,t)\right] \nonumber \\&\quad = \left[ \Pi _h(t) w^n (\cdot ) \nabla \phi (\cdot ,t) \right] (x) + h_k(t) w^n_{k}\psi _{k}(x,t) \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_k(t)+s_1 h_k(t), t) ds_1 ds_0 \nonumber \\&\qquad - h_k(t) w^n_{k+1}\psi _{k+1}(x,t) \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_{k+1}(t) - s_1 h_k(t), t) \, ds_1 ds_0 . \end{aligned}$$
(32)

By using Lemma 1, we substitute (32) into (31), and through a change of variable, we proceed to compute

$$\begin{aligned}&\left| \left( \Pi ^n_h\phi ^n - \Pi ^{n-1}_h\phi ^{n-1} \right) - \int _{t^{n-1}}^{t^n} \left( \Pi _h(t) \frac{\partial \phi }{\partial t} (\cdot ,t) \right) dt \right| \\&\quad = \left| \int _{t^{n-1}}^{t^n}\Pi _h(t) w^n (\cdot ) \nabla \phi (\cdot ,t) (x) + \int _{t^{n-1}}^{t^n}I(x,t) dt \right| \\&\quad \le c \Vert w \Vert _{C^0(L^\infty )} \int _{t^{n-1}}^{t^n} \int _{P_k(t)}^{P_{k+1}(t)} |(\nabla ^2 \phi )(x , t)| dx dt. \end{aligned}$$

By taking the \(L^2\)-norm over \(\Omega \) and applying the Cauchy–Schwartz inequality on the right hand side, we obtain

$$\begin{aligned}&\left\| \left( \Pi ^n_h\phi ^n - \Pi ^{n-1}_h\phi ^{n-1} \right) - \int _{t^{n-1}}^{t^n} \left( \Pi _h(t) \frac{\partial \phi }{\partial t} (\cdot ,t) \right) dt\right\| ^2\nonumber \\&\quad \le c^2 h \Delta t \Vert w \Vert ^2_{C^0(L^\infty )} \int _{\Omega } \int _{t^{n-1}}^{t^n} \int _{K_{{\tilde{k}}}(t)} (\nabla ^2 \phi )(y , t)^2 dy dt dx \nonumber \\&\quad = c^2 h \Delta t \Vert w \Vert ^2_{C^0(L^\infty )} \int _{t^{n-1}}^{t^n} \sum _{k=1}^{N_p-1} h_{k(t)} \int _{K_k(t)} (\nabla ^2 \phi )(y , t)^2 dy dt \nonumber \\&\quad \le c^2 h^2 \Delta t \Vert w \Vert ^2_{C^0(L^\infty )} ||\nabla ^2\phi ||^2_{L^2(t^{n-1},t^n;L^2(\Omega ))}, \end{aligned}$$
(33)

where the dynamic index \({\tilde{k}}={\tilde{k}}(x,t)\) is defined such that \(x\in K_{{\tilde{k}}}(t)\). To complete the proof, we take the square root and divide both sides of (33) by \(\Delta t\), obtaining (16a).

Next, we proof (ii). Therefore, we first rewrite and then further expand the last double integral in (32) as follows

$$\begin{aligned}&\int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_k(t) + (1-s_1) h_k(t), t) ds_1 ds_0\\&= \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_k(t)+s_1 h_k(t), t) ds_1 ds_0\\&\quad + h_k(t) \int _0^1 \int _0^{s_0} \int _{s_1}^{1-s_1} (\nabla ^3 \phi )(P_k(t)+s_2 h_k(t), t) ds_2 ds_1 ds_0. \end{aligned}$$

Additionally we introduce the Taylor expansion

$$\begin{aligned} w^n_{k+1}&= w^n_{k} + h_k \int _0^1 (\nabla w^n)(P_k(t) + sh_k(t)) ds. \end{aligned}$$

By using Lemma 1 and substituting the above expressions into (31) we compute

$$\begin{aligned} A^n(x)&:\;= \frac{1}{\Delta t} \left[ \left( \Pi ^n_h\phi ^n-\Pi ^{n-1}_h\phi ^{n-1} \right) (x) - \int _{t^{n-1}}^{t^n} \left( \Pi _h(t) \frac{\partial \phi }{\partial t} (\cdot ,t) \right) (x) dt \right] \\&= -\frac{1}{\Delta t} \int _{t^{n-1}}^{t^n} h_k(t) w^n_k(\psi _k(x,t) - \psi _{k+1}(x,t)) \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_k(t)+s_1 h_k(t), t) ds_1 ds_0 dt \\&\quad + \frac{1}{\Delta t} \int _{t^{n-1}}^{t^n} h^2_k(t) \psi _{k+1}(x,t) w^n_k \int _0^1 \int _0^{s_0} \int _{s_1}^{1-s_1} (\nabla ^3 \phi )(P_k(t)+s_2 h_k(t), t) ds_2 ds_1 ds_0 dt \\&\quad + \frac{1}{\Delta t} \int _{t^{n-1}}^{t^n} \left( h^2_k(t) \psi _{k+1}(x,t) \int _0^1 (\nabla w^n)(P_k(t) + sh_k(t)) ds \right. \\&\qquad \left. \int _0^1 \int _0^{s_0} (\nabla ^2 \phi )(P_k(t) + (1-s_1) h_k(t), t) ds_1 ds_0\right) dt \\&{=}{:}A^n_1(x) + A^n_2(x) + A^n_3(x). \end{aligned}$$

We proceed by estimating the (\(\Psi ^\prime \))-norm, i.e., \(\Vert A^n\Vert _{\Psi ^\prime }\le \Vert A^n_1\Vert _{\Psi ^\prime } + \Vert A^n_2\Vert _{\Psi ^\prime } + \Vert A^n_3\Vert _{\Psi ^\prime }\). The following bounds hold

$$\begin{aligned} \Vert A^n_1\Vert _{\Psi ^\prime }&\le c_1 h^2 \Vert w\Vert _{C^0(L^\infty )} \Vert \phi \Vert _{H^1(H^3)} \end{aligned}$$
(34)
$$\begin{aligned} \Vert A^n_2\Vert _{\Psi ^\prime }&\le c_2 h^2 \Vert w\Vert _{C^0(L^\infty )} \Vert \phi \Vert _{H^1(H^3)} \end{aligned}$$
(35)
$$\begin{aligned} \Vert A^n_3\Vert _{\Psi ^\prime }&\le c_3 h^2 \Vert w\Vert _{C^0(W^{1,\infty })} \Vert \phi \Vert _{H^1(H^3)} \end{aligned}$$
(36)

and their detailed proofs are provided in Appendix B. Combining all the bounds, we obtain the estimate

$$\begin{aligned} \Vert A\Vert _{\Psi ^\prime }&\le C h^2 \Vert w\Vert _{C^0(W^{1,\infty })} \Vert \phi \Vert _{H^1(H^3)}, \end{aligned}$$

which completes the proof of (16b). \(\square \)

6.1.3 Proof of Corollary 1

We first recall an error estimate for the Lagrange interpolation that follows from [10, Theorem 4.4.20].

Lemma 2

We suppose that Hypothesis 2 holds true and fix \(t\in [0, T]\). Let \(\Pi _h=\Pi _h(t)= \sum _{i=1}^{N_p} f(P_i(t))\psi _i(x,t)\) be the Lagrange interpolation operator at time t and \(v \in H^2(\Omega )\). Then there exists a constant \(C>0\) independent of h such that

$$\begin{aligned} \Vert \Pi _h v - v \Vert _{H^s(\Omega )} \le C h^{2-s} | v |_{H^2(\Omega )} \qquad \text {for }s \in \{0, 1\}. \end{aligned}$$
(37)

Proof of Corollary 1

We show (17a) and assume \(t\in [t^{n-1},t^n]\). By applying the bound (16a) from Theorem 2, Cauchy-Schwartz’s inequality, as well as Lemma 2, we obtain the estimate,

$$\begin{aligned} \left\| \eta ^n - \eta ^{n-1} \right\|&=\left\| (\phi ^n-\Pi ^{n}_h\phi ^n)-(\phi ^{n-1}-\Pi ^{n-1}_h\phi ^{n-1}) \right\| \nonumber \\&=\left\| \phi ^n-\phi ^{n-1} - (\Pi ^n_h\phi ^n-\Pi ^{n-1}_h\phi ^{n-1} ) \right\| \nonumber \\&\le \left\| \int _{t^{n-1}}^{t^n}\left( \frac{\partial \phi }{\partial t} - \Pi _h(t) \frac{\partial \phi }{\partial t} \right) dt \right\| + c_1 h \sqrt{\Delta t} \Vert \phi \Vert _{L^2(t^{n-1},t^n);H^2(\Omega ))} \nonumber \\&\le \sqrt{\Delta t} \sqrt{\int _{t^{n-1}}^{t^n}\left\| \frac{\partial \phi }{\partial t} - \Pi _h \frac{\partial \phi }{\partial t}\right\| ^2_{L^2(\Omega )} dt } + c_1 h \sqrt{\Delta t} \Vert \phi \Vert _{L^2(t^{n-1},t^n);H^2(\Omega ))} \nonumber \\&\le c_2 \sqrt{\Delta t} \left( h^2 \sqrt{\int _{t^{n-1}}^{t^n}\left| \frac{\partial \phi }{\partial t}\right| ^2_{H^2(\Omega )} dt } + h \Vert \phi \Vert _{L^2(t^{n-1},t^n);H^2(\Omega ))} \right) \nonumber \\&\le c_2 h \sqrt{\Delta t} \left( h \Vert \phi \Vert _{H^1(t^{n-1},t^n;H^2(\Omega ))} + \Vert \phi \Vert _{L^2(t^{n-1},t^n);H^2(\Omega ))} \right) . \end{aligned}$$
(38)

To complete the proof, we divide both sides of (38) by \(\Delta t\) and obtain (17a). The bound (17b) is obtained repeating the above estimates in the \(\Psi ^\prime \)-norm, using (16b) instead of (16a) and embedding \(L^2\) in \(\Psi ^\prime \). \(\square \)

6.2 Proof of Theorem 3

To prove Theorem 3, we first state the following lemma.

Lemma 3

(Evaluation of composite functions [20, 32]) 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 (4). Then, the following inequalities hold.

$$\begin{aligned} \Vert \psi \circ X_1(a,\Delta t) \Vert&\le (1+c_1\Delta t) \Vert \psi \Vert , & \forall \psi \in L^2(\Omega ), \end{aligned}$$
(39a)
$$\begin{aligned} \Vert \psi - \psi \circ X_1(a,\Delta t) \Vert&\le c_2\Delta t \Vert \psi \Vert _{H^1(\Omega )}, & \forall \psi \in H^1(\Omega ), \end{aligned}$$
(39b)
$$\begin{aligned} \Vert \psi - \psi \circ X_1(a,\Delta t) \Vert _{H^{-1}(\Omega )}&\le c_3\Delta t \Vert \psi \Vert , & \forall \psi \in L^2(\Omega ). \end{aligned}$$
(39c)

Proof of Theorem 3

We define the terms

$$\begin{aligned} e^n_h {:}{=}\phi ^n_h - \Pi ^n_h \phi ^n, \qquad \eta (t) {:}{=}\phi (t) - \Pi _h (t) \phi (t). \end{aligned}$$

By substituting the error \(e^n_h\) in the numerical scheme (4), we obtain the following expression:

$$\begin{aligned} \left( \frac{e^n_h-[e^{n-1}_h\circ X^n_1]\gamma ^n}{\Delta t},\psi _h\right) +a_0(e^n_h,\psi _h)=\langle R_h^n,\psi _h\rangle , \quad \forall \psi _h\in \Psi ^n_h, \end{aligned}$$
(40)

where the residual on the right hand side is given by

$$\begin{aligned} R^n_h&{:}{=}R^n_1 + R^n_2 + R^n_3, \\ R^n_1&{:}{=}\frac{\partial \phi ^n}{ \partial t} + \nabla \cdot \ (u^n \phi ^n) - \frac{\phi ^n - [\phi ^{n-1}\circ X^n_1]\gamma ^n}{\Delta t}, \\ R^n_2&{:}{=}\frac{\eta ^n - [\eta ^{n-1}\circ X^n_1]\gamma ^n}{\Delta t}, \\ \langle R^n_3, \psi _h\rangle&{:}{=}a_0(\eta ^n,\psi _h). \end{aligned}$$

To obtain an estimate on \(\Vert R_1\Vert \), we follow the error estimate framework for the convection–diffusion problem on a static mesh (details are given in Appendix C.1), which gives us

$$\begin{aligned} \Vert R_1 \Vert _{\ell ^2(\Psi '_h)}&\le c_4 \Delta t \Vert \phi \Vert _{Z^2(0,T)} . \end{aligned}$$
(41)

In case of linear elements in one dimension that are considered here we have \(R_3^n=0\) as is shown in Appendix C.3. To compute a bound for \(R^n_2\) we rewrite it as

$$\begin{aligned} R^n_2&= \frac{\eta ^n - [\eta ^{n-1}\circ X^n_1]\gamma ^n}{\Delta t} \\&= \frac{\eta ^n - \eta ^{n-1}}{\Delta t} + \frac{\eta ^{n-1} - \eta ^{n-1}\circ X^n_1}{\Delta t} + \frac{(\eta ^{n-1} \circ X^n_1 )(1-\gamma ^n)}{\Delta t}. \end{aligned}$$

Then, using (39c) and (39a), noting that thanks to Hypothesis 1 it holds \(1-\gamma ^n \le c_2 \Delta t\), employing (17b), Lemma 2 and embedding \(L^2(\Omega )\) in \(H^1(\Omega )^\prime \), we obtain the following

$$\begin{aligned} \Vert R^n_2 \Vert _{(\Psi ^n_h)'}&\le \left\| \frac{\eta ^n - \eta ^{n-1}}{\Delta t}\right\| _{(\Psi ^n_h)'} + c_3 \Vert \eta ^{n-1} \Vert + c_6 \Vert \eta ^{n-1} \circ X^n_1 \Vert \nonumber \\&\le c_7 \left[ \frac{h^2}{\sqrt{\Delta t}} \Vert \phi \Vert _{H^1(t^{n-1},t^n;H^2(\Omega ))} + h^2 \Vert \phi \Vert _{H^1(H^3)} + \Vert \eta ^{n-1} \Vert \right] \nonumber \\&\le c_8 \left[ \frac{h^2}{\sqrt{\Delta t}} \Vert \phi \Vert _{H^1(t^{n-1},t^n;H^2(\Omega ))} + h^2 \Vert \phi \Vert _{H^1(H^3)} \right] \end{aligned}$$
(42)

Hence, by taking the \(\ell ^2\)-norm

$$\begin{aligned} \Vert R_2\Vert _{\ell ^2(\Psi '_h)}&\le c_9 h^2 \left( \Vert \phi \Vert _{H^1(0,T;H^2(\Omega ))} + \Vert \phi \Vert _{H^1(H^3)} \right) . \end{aligned}$$
(43)

By combining the estimates (41), (43), and taking into account the fact \( R^n_3 =0\), we get

$$\begin{aligned} \Vert R_h \Vert _{\ell ^2(\Psi '_h)} \le C \left( \Delta t \Vert \phi \Vert _{Z^2(0,T)} + h^2 \Vert \phi \Vert _{H^1(0,T;H^2(\Omega ))} + h^2 \Vert \phi \Vert _{H^1(H^3)} \right) , \end{aligned}$$
(44)

as estimate for the total residual, where \(C>0\) is independent of h and \(\Delta t\). Lastly, we apply the stability result from Proposition 2 to problem (40) by substituting \(\phi \) in (19) with \(e^n_h=\phi ^n_h - \Pi ^n_h \phi ^n\), initial value \(e^0_h=0\) and RHS term \(F^n\) as \(R^n_h\). We use the bound (44) to obtained the error estimates (20). \(\square \)

6.3 Proof of Theorem 4

First, we state the following lemma which provides the estimates of the first time step error. The proof is given in Appendix F.

Lemma 4

Suppose that Hypotheses 1, 2, 4, and 3 hold true. Then, it holds that

$$\begin{aligned} ||e^1_h||\le ||e^1_h||+\sqrt{\nu \Delta t}||\nabla e^1_h||\le C (\Delta t^2 +h^2) ||\phi ||_{Z^3\cap H^2(H^2)\cap H^1(H^3)}. \end{aligned}$$
(45)

Proof of Theorem 4

We substitute \(e^n_h\) in the numerical scheme (5) and obtain the following equations for the error:

$$\begin{aligned} \left( \frac{e^n_h-[e^{n-1}_h\circ X^n_1]\gamma ^n}{\Delta t},\psi _h\right) \!+\!a_0(e^n_h,\psi _h)&=\!\langle R_h^n,\psi _h\rangle , \, \forall \psi _h\in \Psi ^n_h, \;n\!=\!1, \end{aligned}$$
(46)
$$\begin{aligned} \left( \frac{ 3 e^n_h \!-\!4e^{n-1}_h \circ X_1^n \gamma ^n \!+\! e^{n-2}_h \circ \tilde{X}_1^n \tilde{\gamma }^n}{2\Delta t} ,\psi _h\right) \!+\!a_0(e^n_h,\psi _h)&=\!\langle \tilde{R}^n_h,\psi _h\rangle , \quad \forall \psi _h\in V^n_h, \; n\ge 2, \end{aligned}$$
(47)

where the residual \(R^n_h,R^n_1,R^n_2,\) and \(R^n_3\) are given as in the proof of Theorem 3, cf., Sect. 6.2, while the residual on the right hand side of (47) is given by:

$$\begin{aligned} \tilde{R}^n_h&{:}{=}\tilde{R}^n_1 + \tilde{R}^n_2 + R^n_3, \\ \tilde{R}^n_1&{:}{=}\frac{\partial \phi ^n}{ \partial t} + \nabla \cdot \ (u^n \phi ^n) - \frac{ 3\phi ^n-4\phi ^{n-1} \circ X_1^n \gamma ^n + \phi ^{n-2} \circ \tilde{X}_1^n \tilde{\gamma }^n}{2\Delta t}, \\ \tilde{R}^n_2&{:}{=}\frac{ 3\eta ^n-4\eta ^{n-1} \circ X_1^n \gamma ^n + \eta ^{n-2} \circ \tilde{X}_1^n \tilde{\gamma }^n}{2\Delta t}. \end{aligned}$$

To obtain an estimate for \( \Vert \tilde{R}_1 \Vert \), we follow the error estimate framework for the general convection–diffusion problem on uniform mesh (details are given in Appendices C.2 and C.3), which gives us

$$\begin{aligned} \Vert \tilde{R}_1 \Vert _{\ell ^2(\Psi '_h)}&\le C_1 \Delta t^2 \Vert \phi \Vert _{Z^3(0,T)} \end{aligned}$$
(48)

and as we have shown in Appendix C.3 it holds \(R_3^n=0\). Next, we compute an estimate for \( \Vert \tilde{R}_2 \Vert _{(\Psi _h^n)^\prime }\). For \(n \ge 2\) it holds

$$\begin{aligned} \Vert \tilde{R}_{2}^n \Vert _{(\Psi _h^n)^\prime }&=\frac{1}{2\Delta t} \Vert 3\eta ^n - 4\eta ^{n-1} \circ X^n_1 \gamma ^n + \eta ^{n-2} \circ \tilde{X}^n_1 \tilde{\gamma }^n \Vert _{(\Psi _h^n)^\prime } \\&=\left\| \frac{3}{2} \bar{D}_{\Delta t } \eta ^n - \frac{1}{2}\bar{D}_{\Delta t} \eta ^{n-1} + \frac{2}{\Delta t}(\eta ^{n-1}-\eta ^{n-1}\circ X^n_1\gamma ^n) - \frac{1}{2\Delta t}(\eta ^{n-2}\right. \\ &\quad \left. -\eta ^{n-2}\circ \tilde{X}^n_1 \tilde{\gamma }^n) \right\| _{(\Psi _h^n)^\prime } \\&\le \frac{3}{2} \Vert \bar{D}_{\Delta t}\eta ^n \Vert + \frac{1}{2} \Vert \bar{D}_{\Delta t}\eta ^{n-1} \Vert + \frac{2}{\Delta t} \Vert (\eta ^{n-1}-\eta ^{n-1}\circ X^n_1\gamma ^n) \Vert _{(\Psi _h^n)^\prime }\\&\quad + \frac{1}{2\Delta t} \Vert (\eta ^{n-2}-\eta ^{n-2}\circ \tilde{X}^n_1 \tilde{\gamma }^n) \Vert _{(\Psi _h^n)^\prime } \\&\le C_3 ( \Vert \bar{D}_{\Delta t}\eta ^n \Vert + \Vert \bar{D}_{\Delta t}\eta ^{n-1} \Vert + \Vert \eta ^{n-1} \Vert + \Vert \eta ^{n-2} \Vert ) \quad (\because \text{( }Lem.3)) \\&\le C_4\left( h^2 \Delta t^{-1/2} \Vert \phi \Vert _{H^1(t^{n-2},t^n;H^2(\Omega ))} + h^2 \Vert \phi \Vert _{H^1(H^3)} \right) , \end{aligned}$$

where the last inequality follows from Lemma 2 and Theorem 2. Taking the \(\ell ^2\)-norm in the previous estimate we obtain

$$\begin{aligned} \Vert \tilde{R}_2 \Vert _{\ell ^2(\Psi '_h)}&\le C_5 h^2 \left( \Vert \phi \Vert _{H^1(0,T;H^2(\Omega ))} + \Vert \phi \Vert _{H^1(H^3)} \right) . \end{aligned}$$
(49)

Combining the bounds (48) and (49) and taking into account the fact \( R_3 =0\), it follows

$$\begin{aligned} \Vert \tilde{R}_h \Vert _{\ell ^2(\Psi '_h)} \!\le \! C ( \Delta t^2 \Vert \!\phi \Vert _{Z^3(0,T)} \!+\! h^2 \Vert \!\phi \Vert _{H^1(0,T;H^2(\Omega ))} \!+\!h^2 \Vert \phi \Vert _{H^1(H^3)} ) , \end{aligned}$$
(50)

where \(C>0\) is independent of h and \(\Delta t\). Finally, from the stability result of Proposition 4 and using \(e^0_h=0\), we get

$$\begin{aligned} ||e_h||_{\ell ^\infty (L^2)} \!+\! \sqrt{\nu } ||\nabla e_h||_{\ell ^2(L^2)}\!&\le \! \left( ||e^1_h|| \!+\! \sqrt{\nu \Delta t} ||\nabla e^1_h||\right) \!+\! \left( ||e_h||_{\ell _2^\infty (L^2)} \!+\! \sqrt{\nu } ||\nabla e^1_h||_{\ell ^2_2(L^2)}\right) \\&\le (||e^1_h|| + \sqrt{\nu \Delta t} ||\nabla e^1_h||) +C||\tilde{R}_h||_{\ell ^2_2(\Psi '_h)} \\&\le C_1 (\Delta t^2 + h^2) ||\phi ||_{Z^3\cap H^2(H^2)\cap H^1(H^3)} \end{aligned}$$

which implies the error estimate (24). We employ Lemma 4 for the estimates of the first time step error such that there is no loss of convergence order. \(\square \)

7 Numerical Experiments

In this section, we present two numerical experiments using the second order LGMM scheme (6) combined with the moving mesh method (12) that show the benefits of the new scheme and verify the error estimate from Theorem 4. As initial data we take \(\phi _h^0=\Pi _h^0 \phi ^0\) from the examples below. To compute the integrals that occur in the scheme we employ the Gauss quadrature of order nine. Since linear finite element spaces are used in our proposed scheme we do not consider higher order quadrature formulae as proposed e.g., in [7]. The linear system appearing in (6) and (12) are iteratively solved using the conjugate gradient (CG) method and successive over-relaxation (SOR) method, respectively. In all experiments we start with an equidistant mesh at the initial time, i.e., for a given \(h_0>0\) the points \(P^0_1,\dots , P_{N_p}^{0}\) are such that

$$\begin{aligned} P_{j+1}^0 - P_j^0 = h_0, \qquad \forall i \in \{1, \dots , N_p \}. \end{aligned}$$
(51)

The numerical results obtained by the new LGMM scheme are compared to analogous results by the LG scheme with static mesh, which can be interpreted as LGMM scheme with points satisfying \(P_j^n=P_j^0\) for all \(i\in \{1,\dots , N_p\}\) and \(n\in \{1,\dots ,N_T\}\) in addition to (51).

Example 1

We consider the domain \(\Omega = (-1,1)\), final time \(T=0.5\) and velocity field \(u(x,t) = 1+\sin (t-x)\) in problem (1). No force field is assumed in this example as we set \(f=0\), and for the boundary conditions we set \(g=0\). We take the initial value \(\phi ^0=\phi (\cdot , 0)\) according to the exact solution

$$\begin{aligned} \phi (x,t) = \exp \left( - \frac{1 - \cos (t-x) }{\nu }\right) . \end{aligned}$$

We solve Example 1 with diffusion coefficient set to \(\nu =0.01\) and \(\nu =0.0001\). In the moving mesh method (12) we set \(\nu _M=\nu \). The integer N determines the discretization of the domain as we choose the initial mesh size \(h_0=2/N\). The time step size is linearly coupled to the initial mesh size through the relation \(\Delta t = 4 h_0\). In this example, since the velocity u does not satisfy Hypothesis 1, i.e.,\(u_{\vert \Gamma } \ne 0\), the non-overlapping condition (cf. Theorem 1) might not be met at the boundary. In this case, we allow the nodal points \(\{P_i^n\}_{i=1}^{N_p}\) to extend beyond the domain.

Table 1 Relative errors and EOCs of LG scheme for Example 1 with \(\nu =0.01\)
Full size table
Table 2 Relative errors and EOCs for of LGMM scheme for Example 1 with \(\nu =0.01\)
Full size table
Table 3 Relative errors and EOCs of LG scheme for Example 1 with \(\nu =0.0001\)
Full size table

In Fig. 1 we show the solution of the LGMM scheme for \(N=512\) and \(\nu =0.01\) in terms of the functions \(\phi ^n_h\) together with the corresponding local mesh or partition sizes \(h_i^n=P_{i+1}^n-P_i^n\) for \(i=1,\dots , N_P\) with respect to their distribution over the computational domain. Clearly, the LGMM scheme maintains a high resolution, i.e., small mesh sizes, in the region, where \(\phi _h\) is large, whereas regions with small \(\phi _h\) are partly significantly lower resolved.

Tables 1, 2, 3 and 4 show the errors and the corresponding experimental orders of convergence (EOC)Footnote 1 of both the LGMM and the LG scheme of second-order after (initial) grid refinement, i.e., iteratively increasing N. In the tables we consider discretization errors with respect to \(L^2(\Omega )\), \(H^1(\Omega )\) and the loss of total mass, defined as:

$$\begin{aligned} E_{\ell ^\infty (L^2)}&{:}{=}\frac{ \Vert \phi _h-\Pi _h\phi \Vert _{\ell ^\infty (L^2)}}{ \Vert \Pi _h\phi \Vert _{\ell ^\infty (L^2)}}, \qquad E_{\ell ^2(H^1_0)}{:}{=}\frac{ \Vert \phi _h-\Pi _h\phi \Vert _{\ell ^2(H^1_0)}}{ \Vert \Pi _h\phi \Vert _{\ell ^2(H^1_0)}}, \quad \\ E_\mathrm{{mass}}&{:}{=}\frac{\left| \int _\Omega \phi ^{N_T}_h dx - \int _\Omega (\Pi _h\phi )^{N_T} dx \right| }{\left| \int _\Omega (\Pi _h\phi )^{N_T} dx\right| }, \end{aligned}$$

where \(\Vert \phi \Vert _{\ell ^2(H_0^1)} {:}{=}\Vert \nabla \phi \Vert _{\ell ^2(L^2})\) and \(\Pi _h\) denotes the time dependent Lagrange interpolation operator at time instance \(t^n\) given as a mapping \(\Pi _h(t^n):C^0(\bar{\Omega })\rightarrow \Psi _h^n\). Due to Theorem 4 and the coupling between \(\Delta t\) and h we expected experimental convergence order 2 in both the \(\ell ^\infty (L^2)\) and the \(\ell ^2(H_0^1)\) (semi-) norm. While the EOCs in the tables mostly support this expectation a decrease in case of higher mesh resolutions for the LG scheme is visible. In the case \(\nu =0.01\) this occurs in \(\ell ^2(H_0^1)\) and becomes more significant also in \(\ell ^\infty (L^2)\) in the case \(\nu =0.0001\). The LGMM scheme does not suffer from this decrease in EOC and provides in the affected cases more accurate numerical solution in terms of both norms. The tables further exhibit a low relative loss of mass as \(E_\mathrm{{mass}}\) is of low magnitude even for coarse grids and further decreases as the mesh is refined. While the mesh movement of the LGMM scheme leads to slightly larger \(E_\mathrm{{mass}}\) on fine meshes in comparison to the LG scheme if \(\nu =0.01\) the loss of mass for the LGMM scheme is significantly smaller than for the LG scheme if \(\nu =0.0001\).

Remark 11

  1. 1.

    Readers might find some of the EOC result of Example 1 for \(N=2048, 4096\) to be unusual. In fact, we observed that the “strange” errors in Example 1 are due to numerical integration errors. In the current computation, we used a numerical integration formula of degree 9. When we use a numerical integration formula of degree 21, we achieve EOCs of approximately 2. Therefore, we can say that our LGMM scheme reduces numerical integration errors, particularly when using high-degree quadrature formulas. We provide a grid convergence study using numerical integration of degree 21 in Appendix G.1.

  2. 2.

    The theoretical analysis does not require the restriction \( g = 0\). We provide an additional example similar to Example 1 with a non-zero boundary condition in Appendix G.2.

Table 4 Relative errors and EOCs of LGMM scheme for Example 1 with \(\nu =0.0001\)
Full size table

Example 2

We consider the domain \(\Omega = (-1,1)\), final time \(T=2\), velocity field \(u(x,t) = \sin (2\pi x)\) and diffusion coefficient \(\nu = 10^{-5}\) in problem (1). Again we take \(f=0\) and \(g=0\). The initial datum is set to \(\phi ^0(x) = \exp [- 100(1 - \cos (x))]\).

We solve Example 2 using scheme (6) combined with the moving mesh method (12), using the parameter \(\nu _M=\nu \), an initial uniform mesh satisfying (51) for \(h_0=2/1024\) and the fixed time step size \(\Delta t = 10^{-4}\), which satisfies condition (13) during the computation. Again the results by the new LGMM scheme are compared to the LG scheme with static mesh. Comparing Figs. 2 and 3, we can observe that while the uniform mesh scheme leads to an oscillating solution, the LGMM scheme is capable to capture the aggregation phenomena. This simulation shows the advantage of the proposed LGMM scheme in capturing sharp spike pattern as observed in bio-medical applications.

Fig. 1
figure 1

Numerical solution \(\phi _h\) and corresponding mesh sizes in Example 1 over the computational domain at time instances \(t = 0\) (left), \(t=0.2340\) (center) and \(t=0.4875\) (right) obtained by the LGMM scheme for \(\nu =0.01\) and \(N=512\)

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Fig. 2
figure 2

Numerical solution \(\phi _h\) and corresponding mesh sizes in Example 2 over the computational domain at time instances \(t = 0\) (left), \(t=1\) (center) and \(t=2\) (right) obtained by the LG scheme with fixed mesh (\(N=256\)). The numerical solution exhibits oscillations

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Fig. 3
figure 3

Numerical solution \(\phi _h\) and corresponding mesh sizes in Example 2 over the computational domain at time instances \(t = 0\) (left), \(t=1\) (center) and \(t=2\) (right) obtained by the LGMM scheme with \(N=256\). The nodal points aggregate along with the solution \(\phi _h\)

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8 Conclusion

In this work, we have equipped the mass-preserving Lagrange–Galerkin scheme of second-order in time with a moving mesh method giving rise to the LGMM scheme, which is capable of numerically solving convection–diffusion problems in one space dimension. We also establish the stability and error estimates of the proposed numerical scheme, the latter being with respect to the \(\ell ^\infty (L^2 )\cap \ell ^2(H_0^1)\) -norm, of order \(O(\Delta t+h^2)\) if the one-step method is used in time and of order \(O(\Delta t^2+h^2)\) if the two-step scheme is used in time. We show numerical results which support the proved error estimates. To this end we have derived a new estimate for the time dependent interpolation operator, which we then embedded in the error estimate framework for the Lagrange–Galerkin method. The numerical simulations also show that the proposed LGMM scheme is capable to capture aggregation phenomena. We believe that the LGMM scheme can be extended to the cases \(d = 2, 3\); though this extension may not be straightforward for all element types used in the spatial discretization. While we anticipate that our method will perform well with P1 triangular elements, more complex elements or those employing higher-order interpolation, such as P2, may require additional considerations or modifications to ensure efficiency of the method. In forthcoming research we consider extensions of our scheme to multidimensional problems as well as applications to real-world problems, especially from biology such as immune system dynamics and cancer growth, in which diffusion and aggregation play crucial roles.