Abstract
This paper presents a multirate in time approach for coupled flow and transport problems combined with goaloriented error control based on the Dual Weighted Residual (DWR) method. The focus is on the implementation of the multirate concept regarding different time scales for the underlying subproblems. Key ingredients are an arbitrary degree discontinuous Galerkin time discretization, an a posteriori error representation for the transport problem coupled with flow and the concept of spacetime slabs based on tensorproduct spaces. From the latter, a spacetime mesh adaptation with highly economical grids is developed. The performance of the approach is studied carefully by numerical convergence examples as well as an example of physical interest for convectiondominated transport.
1 Introduction
In recent years, the coupling of multiphysics and multiscale problems has particulary attracted researchers’ interest; cf., e.g., [1,2,3,4]. Their efficient numerical simulation with respect to the temporal discretization does not become feasible without using different time step sizes adapted to the dynamics and characteristic scales of the respective subproblems. Such methods are referred to as multirate in time (for short, multirate) schemes. For the first time, they were introduced for the numerical approximation of systems of ordinary differential equations in [5, 6]. For a short review of multirate methods including a list of references we refer to [3, 7].
In this work, we focus on the multirate implementation of a fully spacetime adaptive convectiondominated transport problem coupled with a timedependent Stokes flow problem. The implementation is based on our opensource code given in [8], based on the deal.II finite element library; cf. [9]. With regard to our coupled model problem, we assume a highly timedynamic process modeled by the transport equation such that the underlying temporal mesh is discretized using smaller time step sizes compared to a slowly moving process modeled by the viscous flow problem. For instance, such system is used to model species or heat transport in a creeping flow. Beyond that, such multiphysics systems of coupled flow and transport serve as prototype models for applications in several branches of natural and engineering scienes, for instance, contaminant transport and degradation in the subsurface, reservoir simulation, thermal and mass transport in deformable porous media, and many more; cf., e.g., [10,11,12,13,14]. Our motivation is based on the definition of characteristic times measuring different dynamics of the two subproblems. This approach can be traced back to the field of natural sciences and engineering sciences, cf., e.g., [15, 16]
For the sake of physical realism, the transport problem is supposed to be convectiondominated by assuming high Péclet numbers, cf. [17,18,19]. The solutions of such problems are typically characterized by the occurrence of sharp moving fronts and layers. The key challenge for the numerical approximation consists on an accurate and efficient solution while avoiding nonphysical oscillations or smearing effects. The application of stabilization techniques is a typical approach to overcome these effects. As shown in a comparative study for convectiondominated problems in [20], stabilization techniques on globally refined meshes fail to avoid these oscillations even after tuning stabilization parameters. Thus, for an efficient numerical simulation of multiphysics problems handling the challenges described above, it is indisputable that adaptive mesh refinement strategies in space and time are necessary. One possible technique for those adaptive strategies is goaloriented a posteriori error control based on the Dual Weighted Residual method [21, 22]. For a general review of a posteriori error estimation we refer to [23, 24].
In this work, we extend our approaches and implementations from [8, 25] considering a timedependent flow problem that needs to be solved on a different time scale than the transport problem. Precisely, the features of this work are the following:

Development of a multirate concept with independent time scales for the transport and flow problem, respectively.

Implementation of tensorproduct spacetime slabs for an arbitrary order discontinuous Galerkin (dG) time discretization.

Implementation of coupling the Stokes flow velocity to the transport problem using interpolation techniques between different finite element spaces and meshes.
This work is organized as follows. In Sect. 2 we introduce the model problem, the multirate decoupling of the transport and flow problems and their spacetime discretizations. In Sect. 3 we derive an a posteriori error representation for the transport problem. In Sect. 4 we explain the implementation of the spacetime tensorproduct spaces. The underlying algorithm and some related aspects are presented in Sect. 5. Numerical examples are given in Sects. 6 and 7 we summarize with conclusions and give some outlook for future work.
2 Model Problem, Multirate and SpaceTime Discretization
In this section, we introduce the coupled model problem, explain our multirate in time approach and present some details of the spacetime discretizations.
2.1 Model Problem
The time dependent convection–diffusion–reaction transport problem in dimensionless form is given by
where \(\varOmega \subset {\mathbb {R}}^{d}\), with \(d=2,3\), is a polygonal or polyhedral bounded domain including a boundary partition \(\partial \varOmega = \varGamma _D \cup \varGamma _N\), \(\varGamma _D \ne \emptyset \) with outer unit normal vector \(\mathbf{n}\). In addition, \(I=(0,T]\), \(0< T < \infty \), is a finite time interval. The characteristic time \(t_{\text {transport}}\) of this transport equation (1) can be understood as a dimensionless time variable depending on the diffusive, convective as well as reactive part and is here defined by
where \(0 < \varepsilon \ll 1\) is the diffusion coefficient, \(\alpha > 0\) is the reaction coefficient, L denotes the characteristic length of the domain \(\varOmega \), for instance, its diameter, and V denotes a characteristic velocity of the flow field \(\mathbf{v}\), for instance, the mean inflow velocity given by \(\frac{1}{T\, \varGamma _{\mathrm {in}}} \int _I \int _{\varGamma _{\mathrm {in}}} \mathbf{v}_D\cdot ( \mathbf{n}) \;\mathrm {d} o\; \mathrm {d} t\); cf. [15, 16] for more details.
The convection field \(\mathbf{v}\) in the transport problem (1) is determined by the dimensionless Stokes flow system
for a boundary partition \(\partial \varOmega = \varGamma _{\text {in}} \cup \varGamma _{\text {wall}} \cup \varGamma _{\text {out}}\) which is (in general) independent from the boundary partition of the transport problem. The appropriate choice for the boundary partition and setting of the inflow profiles is standard and can be found in the literature [26]. We assume that \({\nu }> 0\) is a viscosity coefficient. Further, \(\varvec{\epsilon }(\mathbf{v}):=(\nabla \mathbf{v}+(\nabla \mathbf{v})^\top )/2\) is the strain tensor. The characteristic time \(t_{\text {flow}}\) of the Stokes flow equation is then defined by
with L and V being chosen as in (2). With regard to the characteristic times of the two subproblems, we assume that \(t_{\text {transport}} \ll t_{\text {flow}}\) such that we are using a finer temporal mesh to resolve the dynamics of a faster process given by the transport equation compared to the slower process of the viscous, creeping flow. This multirate in time approach is described in detail in the following section.
Wellposedness of (1), (3) and the existence of a sufficiently regular solution, such that all of the arguments and terms used below are welldefined, are tacitly assumed without mentioning all technical assumptions about the data and coefficients explicitly, cf. [27] and [28].
2.2 Multirate
For an efficient approximation, we use a multirate in time approach to mimic the behaviour of a slowly moving fluid, that is approximated by a timedependent Stokes solver, and a faster convection–diffusion–reaction process. Precisely, the problems given by (1) and (3) are considered on different time scales modeling the underlying physical processes. We initialize the temporal mesh independently for the Stokes flow and the transport problem with the following properties

the Stokes flow temporal mesh is not finer than that of the transport problem,

the endpoints of the temporal mesh of the Stokes solver must match with endpoints of the temporal mesh of the transport problem.
We allow for adaptive time refinements within the transport problem and for global time refinements within the Stokes solver. The latter is due to the lack at the time being of an error estimator for the current implementation of the Stokes flow solver. An exemplary initialization and one manufactured refined temporal mesh are illustrated in Fig. 1.
For the multirate decoupling of the transport problem, we let \(0 =: t_0^\text {}< \dots < t_{N^\ell }^\text {} := T\) a set of time points for a partition of the time domain I into leftopen subintervals \(I_n:=(t_{n1}^\text {},t_n^\text {}]\), \(n=1,\dots ,N^\ell \). The number \(N^\ell \) depends on the adaptivity loop \(\ell \). In the same way, but in accordance with allowing for a different temporal mesh of the Stokes flow problem, we introduce the following set of time points \(0 =: t_0^\mathrm {f}< \dots < t_{N^{\mathrm {f},\ell }}^{\mathrm {f}} := T\), using an index \(\mathrm {f}\) when dealing with the flow problem here and in the following. We approximate the solution \(\{\mathbf{v}, {p}\}\) of the Stokes flow problem on each \(I_n^\mathrm {f}\) by means of a piecewise constant discontinuous Galerkin (dG(0)) time approximation. For simplicity of the implementation, we ensure that each element of the set \(\{ t_0^\mathrm {f}, \dots , t_{N^{\mathrm {f},\ell }}^\mathrm {f} \}\) corresponds to an element of the set \(\{ t_0^\text {}, \dots , t_{N^\ell } \}\). Additionally, we approximate the solution of the transport problem with an arbitrary degree \(r \ge 0\) in time. This gives us for \(r>0\) an additional level of the multirate in time character between the two problems.
2.3 Weak Formulation
In this section, we present the weak formulation of the transport and Stokes flow problem given by (1) and (3), respectively, to prepare the discretizations in space and time following below. Let \(X:=\{u \in L^2(0, T; H^1_0(\varOmega )) \mid \partial _t u \in L^2(0, T; H^{1}(\varOmega ))\}\) and \(Y_1 := \{ {\varvec{v}} \in L^2(0, T; H^1_0(\varOmega )^d) \mid \partial _t {\varvec{v}} \in L^2(0, T; H^{1}(\varOmega )^d) \}\). Then, the weak formulation of (1) reads as follows:
For a given \(\mathbf{v}\in Y_1\) of (6), find \({u}\in X\) such that
where the bilinear form \(A: \{X;Y_1\} \times X \rightarrow {\mathbb {R}}\) and the linear form \(G: L^2(0, T;\) \(H^{1}(\varOmega )) \rightarrow {\mathbb {R}}\) are defined by
Here, \((\cdot , \cdot )\) denotes the inner product of \(L^2(\varOmega )\) or duality pairing of \(H^{1}(\varOmega )\) with \(H^1_0(\varOmega )\), respectively. By \(\Vert \cdot \Vert \) we denote the associated \(L^2\)norm.
For the weak formulation of (3) we additionally define \(Y_2 := \{p \in L^2(0,T;L^2(\varOmega ))\}\). Then we get:
For \(\mathbf{f}\in L^2(I;H^{1}(\varOmega )^d)\) and \(\mathbf{v}_0 \in L^2(\varOmega )^d\), find \(\{ \mathbf{v}, {p}\} \in Y_1 \times Y_2\), such that
where the bilinear form \(B: \{ Y_1 \times Y_2 \} \times \{ Y_1 \times Y_2 \} \rightarrow {\mathbb {R}}\) and the linear form \(F: L^2(0,T;H^{1}(\varOmega )^d) \rightarrow {\mathbb {R}}\) are defined by
2.4 Discretization in Time
The sets of time subintervals \(I_n\) and \(I_n^\mathrm {f}\) as introduced in Sect. 2.2 are finite and countable. Therefore, the separation of the global spacetime cylinder \(Q=\varOmega \times I\) into a partition of spacetime slabs \({\hat{Q}}_n = \varOmega \times I_n\) for the transport problem and \({\hat{Q}}_n^\mathrm {f} = \varOmega \times I_n^\mathrm {f}\) for the Stokes flow problem, respectively, is reasonable. The time domain of each spacetime slab \({\hat{Q}}_n\) or \({\hat{Q}}_n^\mathrm {f}\) is then discretized using a onedimensional triangulation \({\mathcal {T}}_{\tau ,n}\) or \({\mathcal {T}}_{\sigma ,n}\) for the subinterval \(I_n\) or \(I_n^\mathrm {f}\), respectively. This allows to have more than one cell in time on a slab \({\hat{Q}}_n\) or \({\hat{Q}}_n^\mathrm {f}\) and a different number of cells in time of pairwise different slabs \({\hat{Q}}_i\) and \({\hat{Q}}_j\) or \({\hat{Q}}_i^\mathrm {f}\) and \({\hat{Q}}_j^\mathrm {f}\), \(1 \le i,j \le N^\ell , N^{\mathrm {f},\ell }\). Furthermore, let \({\mathcal {F}}_\tau \) and \({\mathcal {F}}_\sigma \) be the sets of all interior time points given as
The commonly used time step size \(\tau _K\) or \(\sigma _K\) is here the diameter or length of the cell in time \(K_n\) of \({\mathcal {T}}_{\tau ,n}\) or \(K_n^\mathrm {f}\) of \({\mathcal {T}}_{\sigma ,n}\) and the global time discretization parameter \(\tau \) or \(\sigma \) is the maximum time step size \(\tau _K\) or \(\sigma _K\) of all cells in time of all slabs \({\hat{Q}}_n\) or \({\hat{Q}}_n^\mathrm {f}\).
For the discretization in time of the transport problem (5) we use a discontinuous Galerkin method dG(r) with an arbitrary polynomial degree \(r \ge 0\). Let \(X_{\tau }^{r}\) be the timediscrete function space given as
where \({\mathcal {P}}_{r}(K_n; H_0^1(\varOmega ))\) denotes the space of all polynomials in time up to degree \(r \ge 0\) on \(K_n\) with values in \(H_0^1(\varOmega )\) . For some discontinuous in time function \(u_{\tau } \in X_{\tau }^{r}\) we define the limits \(u_{\tau }(t_F^\pm )\) from above and below as well as their jump at \(t_F\) by
The semidiscretization in time of the the transport problem (5) then reads as follows:
For a given \(\mathbf{v}_{\sigma } \in Y_{\sigma }^{r}\) of (11), find \({u}_\tau \in X_{\tau }^{r}\) such that
where the semidiscrete bilinear form and linear form are given by
with the bilinear form \(a(\cdot ,\cdot )(\cdot )\) depending on the semidiscrete Stokes solution \(\mathbf{v}_\sigma \).
The discontinuous timediscrete function space for the flow problem is given by
Then, the semidiscretization in time of the the Stokes flow problem (6) reads as:
Find \(\{ \mathbf{v}_{\sigma },{p}_\sigma \} \in Y_{\sigma }^{r}\) such that
where the semidiscrete bilinear form and linear form are given by
2.5 Discretization in Space and SUPG Stabilization
Next, we describe the Galerkin finite element approximation in space of the semidiscrete transport problem (8) and the flow problem (11), respectively. We use Lagrange type finite element spaces of continuous functions that are piecewise polynomials. For the discretization in space, we consider a separation \(Q_n={\mathcal {T}}_{h,n}\times I_n\) or \(Q_n^{\mathrm {f}}={\mathcal {T}}_{h,n}^{\mathrm {f}}\times I_n^{\mathrm {f}}\), where \({\mathcal {T}}_{h,n}\) or \({\mathcal {T}}_{h,n}^{\mathrm {f}}\) build a decomposition of the domain \(\varOmega \) into disjoint elements K or \(K^{\mathrm {f}}\), such that \({\overline{\varOmega }}=\cup _{K\in {\mathcal {T}}_{h}}{\overline{K}}\) or \({\overline{\varOmega }}= \cup _{K^{\mathrm {f}}\in {\mathcal {T}}_{h}^{\mathrm {f}}}\overline{K^{\mathrm {f}}}\) for the transport and Stokes flow problem, respectively. Here, we choose the elements \(K\in {\mathcal {T}}_{h}\) or \(K^{\mathrm {f}}\in {\mathcal {T}}_{h}^{\mathrm {f}}\) to be quadrilaterals for \(d=2\) and hexahedrals for \(d=3\). We denote by \(h_{K}\) or \(h_{K}^{\mathrm {f}}\) the diameter of the element K or \(K^{\mathrm {f}}\). The global space discretization parameter h or \(h^{\mathrm {f}}\) is given by \(h:=\max _{K\in {\mathcal {T}}_{h}}h_{K}\) or \(h^{\mathrm {f}}:=\max _{K^{\mathrm {f}}\in {\mathcal {T}}_{h}^{\mathrm {f}}}h_{K}^{\mathrm {f}}\), respectively. Our mesh adaptation process yields locally refined cells, which is enabled by using hanging nodes. We point out that the global conformity of the finite element approach is preserved since the unknowns at such hanging nodes are eliminated by interpolation between the neighboring ’regular’ nodes; cf. [22, Chapter 4.2] and [29] for more details. On \({\mathcal {T}}_{h}\) and \({\mathcal {T}}_{h}^{\mathrm {f}}\) we define the discrete finite element spaces by \( V_{h}^{p,n}:= \big \{v\in C(\overline{\varOmega })\mid v_{K} \in Q_h^p(K)\,,\forall K\in \mathcal {T}_{h},n \big \}\,, \) and \( V_{h^{\mathrm {f}}}^{p,n}:= \big \{v\in C(\overline{\varOmega })\mid v_{K^{\mathrm {f}}} \in Q_h^p(K^{\mathrm {f}})\,,\forall K^{\mathrm {f}}\in \mathcal {T}_{h}^{\mathrm {f}},n \big \}\,, \) with \(1 \le n \le N,N^{\mathrm {f}}\), where \(Q_h^p(K)\) or \(Q_{h^{\mathrm {f}}}^p(K^{\mathrm {f}})\) is the space defined on the reference element with maximum degree p in each variable. By replacing \(H_0^1(\varOmega )\) in the definition of the semidiscrete function space \(X_{\tau }^{r}\) in (7) by \(V_h^{p,n}\) and by replacing \(H_0^1(\varOmega )^d, L^2(\varOmega )\) in the definition of the semidiscrete function space \(Y_{\sigma }^{r}\) in (10) by \(V_{h^{\mathrm {f}}}^{p,n}\), we obtain the fully discrete function spaces for the transport and Stokes flow problem, respectively,
We note that the spatial finite element space \(V_h^{p,n}\) and \(V_{h^{\mathrm {f}}}^{p,n}\) are allowed to be different on all subintervals \(I_n\) and \(I_n^{\mathrm {f}}\), respectively, which is natural in the context of a discontinuous Galerkin approximation of the time variable and allows dynamic mesh changes in time. Due to the conformity of \(H_h^{p_{{u}},n}\), \(H_h^{p_{\mathbf{v}},n}\) and \(L_h^{p_{{p}},n}\), we get \(X_{\tau h}^{r,p}\subseteq X_{\tau }^{r}\) and \(Y_{\sigma h^{\mathrm {f}}}^{r,p}\subseteq Y_{\sigma }^{r}\), respectively.
For convectiondominated transport, the finite element approximation needs to be stabilized in order to avoid spurious and nonphysical oscillations of the discrete solution arising close to sharp fronts and layers. Here, we apply the streamline upwind Petrov–Galerkin (SUPG) method introduced by Hughes and Brooks [30, 31]. With this in mind,the stabilized fully discrete discontinuous in time scheme for the transport problem reads as follows:
For a given \(\mathbf{v}_{\sigma h} \in Y_{\sigma h^{\mathrm {f}}}^{r,p}\) of (15), find \({u}_{\tau h} \in X_{\tau h}^{r,p}\) such that
where the linear form \(G_{\tau }(\cdot )\) is defined in (9) and the stabilized bilinear form \(A_{S}(\cdot ;\cdot )(\cdot )\) is given by
with \(A_{\tau }(\cdot ;\cdot )(\cdot )\) being defined in (9). Here, \(S_A(\cdot ;\cdot )(\cdot )\) is the SUPG stabilized bilinear form obtained by adding weighted residuals. In order to keep this work short, we skip here the explicit presentation of \(S_A\) that can be found, e.g., in our work [25, Sec. 2.5].
Finally, the fully discrete discontinuous in time scheme for the Stokes flow problem reads as follows:
Find \(\{ \mathbf{v}_{\sigma h},{p}_{\sigma h} \} \in Y_{\sigma h^{\mathrm {f}}}^{r,p}\) such that
with \(B_{\sigma }(\cdot ,\cdot )(\cdot ,\cdot )\) and \(F_{\sigma }(\cdot )\) being defined in (12).
3 An A Posteriori Error Estimator for the Transport Problem
In this section, we present a DWRbased a posteriori error representation for the stabilized transport problem (14) coupled with the flow problem via the convection tensor \(\mathbf{v}_{\sigma h}\) given by (15). The following error representation formulas are given in terms of a user chosen goal functional \(J(\cdot )\) by using Lagrangian functionals \({\mathcal {L}}\) within a constrained optimization approach, cf. [21, 22, 32]. Since the derivation is close to our work based on a coupling of a steadystate Stokes problem, we keep this section rather short by presenting only the main result regarding the separation of the temporal and spatial discretization errors that serve as local error indicators for the adaptive mesh refinement. For a detailed derivation and more details, we refer to our work [25].
Theorem 1
Let \(\{{u},{z}\}\in X \times X\), \(\{{u}_{\tau },{z}_{\tau }\} \in X_{\tau }^{r} \times X_{\tau }^{r}\), and \(\{{u}_{\tau h},{z}_{\tau h}\} \in X_{\tau h}^{r,p} \times X_{\tau h}^{r,p}\) denote stationary points of \({\mathcal {L}}, {\mathcal {L}}_{\tau }\), and \({\mathcal {L}}_{\tau h}\) on different discretization levels, i.e.,
Additionally, for the errors \(e = {u} {u}_{\tau }\) and \(e = {u}_{\tau }  {u}_{\tau h}\) we have the Eqs. (27) and (28) of Galerkin orthogonality type. Then, for the discretization errors in space and time we get the representation formulas
with nonvanishing Galerkin orthogonality terms \({\mathcal {D}}_{\tau }(\cdot ,\cdot )\) and \({\mathcal {D}}_{\tau h}(\cdot ,\cdot )\), given by
where \({\mathcal {D}}^\prime _{\tau }(\cdot ,\cdot )(\cdot ,\cdot )\) and \({\mathcal {D}}^\prime _{\tau h}(\cdot ,\cdot )(\cdot ,\cdot )\) denote the Gâteaux derivatives with respect to the first and second argument. Here, \(\{{\tilde{{u}}}_{\tau },{\tilde{{z}}}_{\tau }\}\in X_{\tau }^{r} \times X_{\tau }^{r}\), and \(\{{\tilde{{u}}}_{\tau h},{\tilde{{z}}}_{\tau h}\} \in X_{\tau h}^{r,p} \times X_{\tau h}^{r,p}\) can be chosen arbitrarily and the remainder terms \({\mathcal {R}}_{\tau }, {\mathcal {R}}_{h}\) are of higherorder with respect to the errors \({u}{u}_\tau ,{z}{z}_\tau \) and \({u}_{\tau }{u}_{\tau h},{z}_{\tau }{z}_{\tau h}\), respectively.
The explicit presentations of the Lagrangian functionals \({\mathcal {L}}, {\mathcal {L}}_{\tau }\) and \({\mathcal {L}}_{\tau h}\) as well as the primal and dual residuals based on the continuous and semidiscrete schemes \(\rho , \rho ^{*}\) and \(\rho _\tau ,\rho _\tau ^*\), respectively, are given in the Appendix.
Remark 1
We note that within the temporal error representation formula (first equation in (16)) additional terms due to the coupling occur, cf. Eq. (17). This is an extension to our previous results obtained in [33] and [25]. Furthermore, we indicate that the occurring differences \(v{\tilde{v}}_{\tau }\) and \(v_{\tau }{\tilde{v}}_{\tau h}\) with regard to the primal and dual variables are called temporal and spatial weights, respectively.
Proof
The technique to prove the temporal error representation formula (first equation in (16)) is equivalent to the spatial counterpart that can be found in our work [25, Thm. 3.1] and was originally proved by Besier and Rannacher applied to the incompressible NavierStokes equations in [32, Thm. 5.2]. More precisely, we are using a general result given in [32, Lemma 5.1] with the following settings:
where \({\mathcal {L}},{\mathcal {L}}_{\tau }\) are the Lagrangian functionals given by (29) and \(Y,Y_1\) and \(Y_2\) are function spaces defined in [32, Lemma 5.1]. \(\square \)
4 Implementation of TensorProduct Spaces
In this section, we explain the implementation of spacetime tensorproduct spaces in detail. We restrict the explanation to the case of the scalarvalued transport equation with primal and dual finite element spaces. The implementation for the primal vectorvalued Stokes flow problem is very similar with the difference that the spatial finite element has d+1 components for the velocity and pressure variables. We denote the number of spatial degrees of freedom by \(N_{\mathrm {DoF}}^{\mathrm {s,n}}\) for one degree of freedom in time and the number of temporal degrees of freedom by \(N_{\mathrm {DoF}}^{\mathrm {t,n}}\) on the nth slab.
To implement the spacetime tensorproduct space, as illustrated in Fig. 2, we start with the usual discretization of the finite element method in space having only one degree of freedom in time in an adaptive time marching process, but here we do this for each slab. Therefore, we generate the geometrical triangulation, i.e. a spatial mesh, and colourize the boundaries. Boundary colours can mark for instance Dirichlet type boundary conditions, Neumann type boundary conditions, etc. Next, we initialize each slab by creating an independent copy of the generated spatial triangulation.
Then, for one degree of freedom in time on each slab, we distribute the spatial degrees of freedom and generate affine constraints objects. Remark that an affine constraints object may include information on handling degrees of freedom on hanging nodes or on Dirichlet type boundary nodes. The sparsity pattern for a sparse matrix is now generated with the geometric triangulation, the spatial degree of freedom (DoF) handler and the constraints object for one degree of freedom in time.
Next, the spacetime tensorproduct degrees of freedom on a slab are aligned by their local degree of freedom in time on a slab. Precisely, the first degree of freedom in time has the global number 0 and the last one has the number \(N_{\mathrm {DoF}}^{\mathrm {t,n}}1\). The numbering of the local temporal degrees of freedom is increasingly ordered by their mesh cell index. Remark that we have an additional onedimensional triangulation (temporal mesh) for the time subinterval \((t_{n1}, t_n)\) corresponding to the nth slab; refer to the Fig. 3 for details. Overall, we have \(N_{\mathrm {DoF}}^{\mathrm {t,n}}\) times \(N_{\mathrm {DoF}}^{\mathrm {s,n}}\) degrees of freedom on the nth slab. Then, the spacetime tensorproduct constraints are created by taking the original constraints object and shifting all entries accordingly such that the \(N_{\mathrm {DoF}}^{\mathrm {t,n}}\) are represented. Precisely, the spatial degrees of freedom from 0 to \(N_{\mathrm {DoF}}^{\mathrm {s}}1\) are associated to the first local temporal degree of freedom on a slab. If there are more than one temporal degrees of freedom on a slab, the corresponding spatial degrees of freedom are shifted by the number \(N_{\mathrm {DoF}}^{\mathrm {s,n}}\) times the local temporal degree of freedom index.
For each degree of freedom in time, the sparsity pattern is now copied into the diagonal blocks for the spacetime tensor product sparsity pattern. A higherorder polynomial degree in time introduces couplings between the temporal basis functions resulting in additional coupling blocks. For the case of more than one time cell per slab, additional couplings appear for temporal derivatives between the time basis functions of two consecutive time cells. For the primal problem, the evolution is forward in time and therefore these couplings appear in the left lower part. For the dual problem, the evolution is backward in time and therefore the coupling diagonals appear in the right upper part. Exemplary sparsity patterns are given in Fig. 4.
For the assembly process we can use the basis functions and their derivatives in time similar to the classical finite element approach in space. But the distribution of the local contributions must respect the order of the temporal basis functions. First, we take the mapping from a local to a global degree of freedom in space. To respect the temporal basis functions, we shift the local to global mapping accordingly by the factor of local degrees of freedom in space on a spatial cell. This results in a shift of each global degree of freedom by the factor of \(N_{\mathrm {DoF}}^{\mathrm {s,n}}\) times the global degree of freedom of the respective basis function in time. The local matrix has therefore the size of the local degrees of freedom on a spatial mesh cell times the local degrees of freedom in time on a temporal mesh cell. In the case of more than on time cell per slab, an additional local matrix is assembled for the coupling of the trial basis functions of the previous time cell and the test basis functions of the current time cell. This implements the negative part of the jump trace operator in time which is transferred to the righthand side in a classical time marching approach.
Finally, the spacetime constraints of the slab have to be applied to the system matrix, the solution vector and the righthand side vector. The spacetime hanging node constraints have to be condensed in the solution vector after solving the linear system for all degrees of freedom on the slab.
5 Algorithm
Our spacetime adaptivity and mesh refinement strategy uses the following algorithm. Here, we are interested to control the transport problem under the condition that the influence of the error in the Stokes flow problem stays small.
Remark 2

For the spatial discretization of the Stokes flow problem we are using TaylorHood elements \(Q_p/Q_{p1}\,, p \ge 2\). In order to ensure the conditions to the temporal meshes outlined in Sect. 2.2, we refine the transport meshes after the Stokes flow meshes such that the endpoints in the temporal mesh of the Stokes solver match with endpoints in the temporal mesh of the transport solver.

Within the Steps 2, 4 and 5 of the algorithm, the computed convection field \(\mathbf{v}_{\sigma h}\) of the Stokes problem is interpolated to the adaptively refined spatial and temporal triangulation of the spacetime slabs. By means of the variational equation (15) along with the definition (12) of the bilinear form \(B_{\sigma }\) the fully discrete velocity \(\mathbf{v}_{\sigma h}\) is weakly divergence free on the spatial mesh \({\mathcal {T}}_{\sigma ,n}\) of the flow problem. However, on the spatial mesh \({\mathcal {T}}_{\tau ,n}\) of the transport problem this constraint is in general violated due to the different decomposition of \(\varOmega \). The lack of divergencefreeness might be the source of an approximation error. By the application of an additional Helmholtz or Stokes projection (cf. [34, Rem. 2.1]) the divergencefree constraint can be recovered on the spatial mesh of the transport quantity \(u_{\tau h}\). This amounts to the solution of a problem of Stokes type for each of the fully discrete velocities fields of the temporal flow mesh (cf. Fig. 1). In the numerical experiments of Sect. 6 we did not observe any problems without postproccesing the velocity fields to ensure discrete divergencefreeness on \({\mathcal {T}}_{\tau ,n}\).

Our simulation tools of the DTM++ project are frontend solvers for the deal.II library; cf. [9]. Technical details of the implementation are given in [8, 25].
In the following, we give some details regarding the localization of the error representations derived in Theorem 1. Their practical realization and the definition of error indicators \(\eta _{\tau }\) and \(\eta _{h}\) are obtained by neglecting the remainder terms \({\mathcal {R}}_{\tau }\) and \({\mathcal {R}}_{h}\) and splitting the resulting quantities into elementwise contributions by means of the classical approach using integration by parts on every single mesh element, cf. [21].
To compute these error indicators, we replace all unknown solutions by the approximated fully discrete solutions \({u}_{\tau h} \in X_{\tau h}^{r, p}\), \({z}_{\tau h} \in X_{\tau h}^{r, q}\), with \(p < q\), and \(\mathbf{v}_{\sigma h} \in Y_{\sigma h^{\mathrm {f}}}^{0,p_\mathbf{v}}\), \(p_\mathbf{v}\ge 2\), whereby the arising weights are approximated in the following way.

Approximate the temporal weights \(u{\tilde{u}}_{\tau }, z{\tilde{z}}_{\tau }\) by means of a higherorder reconstruction using GaussLobatto quadrature points, exemplary given by
$$\begin{aligned} \begin{array}{rcl} u{\tilde{u}}_{\tau }\approx & {} \mathrm {E}_{\tau }^{(r+1)}u_{\tau h}u_{\tau h}\,, \end{array} \end{aligned}$$using an reconstruction in time operator \(\mathrm {E}_{\tau }^{(r+1)}\) thats acts on a time cell of length \(\tau _K\) and lifts the solution to a piecewise polynomial of degree (r+1) in time. This approximation technique is a new approach compared to out previous work [25, Sec. 4], where a higherorder finite element approximation was used, and is done for the purpose to reduce numerical costs solving the dual problem.

Approximate the spatial weights \(u_{\tau }{\tilde{u}}_{\tau h}\) and \(z_{\tau }{\tilde{z}}_{\tau h}\) by means of a patchwise higherorder interpolation and a higherorder finite elements approach, respectively, given by
$$\begin{aligned} \begin{array}{rcl} u_{\tau }{\tilde{u}}_{\tau h} &{} \approx &{} \mathrm {I}_{2h}^{(2p)}u_{\tau h}u_{\tau h}\,, \\ z_{\tau }{\tilde{z}}_{\tau h} &{} \approx &{} z_{\tau h}\mathrm {R}_{h}^{p}z_{\tau h}\,, \end{array} \end{aligned}$$using an interpolation in space operator \(\mathrm {I}_{2h}^{(2p)}\) and an restriction in space operator \(\mathrm {R}_{h}^{p}\) that are described in detail in our work [25, Sec. 4].
6 Numerical Examples
In this section, we study the convergence, computational efficiency and stability of the introduced DWRbased adaptivity approach for the coupled transport and Stokes flow problem. The first example given in Sect. 6.1 is an academic test problem with given analytical solutions to study the convergence behavior of the two subproblems and, in particular, the coupling between them. The second example given in Sect. 6.2 serves to demonstrate the performance properties of the algorithm with regard to adaptive mesh refinement in space and time. Finally, the third example in Sect. 6.3 is motivated by a problem of physical relevance in which we simulate a convectiondominated transport with goaloriented adaptivity of a species through a channel with a constraint.
6.1 Example 1 (HigherOrder SpaceTime Convergence Studies)
In a first numerical example, we study the spacetime higherorder convergence behavior to validate the correctness of the higherorder implementations in space and time. Therefore, we consider the two cases of a solely solved Stokes flow problem as well as a nonstabilized solved convection–diffusion–reaction transport problem coupled with this Stokes equation via the convection field solution \(\mathbf{v}_{\sigma h}\). The latter may be compared to the results of a solely solved transport equation with a constant convection field \(\mathbf{v}=(2,3)^\top \) published in our work [25, Example 1]. For this purpose, we investigate problem (3) with the given analytical solution
with \({\varvec{x}} = (x_1, x_2)^\top \in {\mathbb {R}}^2\,, t \in {\mathbb {R}}\) and \(\nabla \cdot \mathbf{v}= 0\). The viscosity is set to \({\nu }=0.5\). The problem is defined on \(Q=\varOmega \times I:=(0,1)^2\times (0,1]\). The initial and boundary conditions are given as
and the volume force term \(\mathbf{f}\) is calculated from the given analytical solution (20) and Eq. (3). This example is a typical test problem for timedependent incompressible flow and can be found, for instance, in [32, Example 1].
For the following test settings, the solution \(\{\mathbf{v},{p}\}\) is approximated with the spacetime higherorder methods {cG(2)–dG(2), cG(1)–dG(2)} and {cG(3)–dG(3), cG(2)–dG(3)}, respectively. Due to the same polynomial orders of the spatial and temporal discretizations with respect to the convection field \(\mathbf{v}\), and lower polynomial order in space compared to in time with respect to the pressure variable \({p}\), we expect experimental orders of convergence (EOC \(:= \log _2( e _\ell /  e _{\ell 1})\)) for the convection field \(\mathbf{v}\) of \(\text {EOC}^{2,2} \approx 3\) for the cG(2)–dG(2) method and \(\text {EOC}^{3,3} \approx 4\) for the cG(3)–dG(3) method, as well as experimental orders of convergence for the pressure variable \({p}\) of \(\text {EOC}^{1,2} \approx 2\) for the cG(1)–dG(2) method and \(\text {EOC}^{2,3} \approx 3\) for the cG(2)–dG(3) method for a global refinement convergence test. The results are given by Table 1 and nicely confirm our expected results for the respective spatial and temporal discretizations.
The second part of the first example now serves to verify the higherorder implementation of the coupled problem. Therefore, the convection–diffusion–reaction transport problem (1) is coupled with the Stokes flow problem (3) via the convection field solution \(\mathbf{v}_{\sigma h}\), using the exact solution \(\mathbf{v}\) given by Eq. (20). More precisely, we study problem (1) with the given analytical solution
with \(m_1(t) := \frac{1}{2} + \frac{1}{4} \cos (2 \pi t)\) and \(m_2(t) := \frac{1}{2} + \frac{1}{4} \sin (2 \pi t)\), and, \(\nu _1({{\hat{t}}}) := 1\), \(\nu _2({{\hat{t}}}) := 5 \pi \cdot (4 {{\hat{t}}}  1)\), for \({{\hat{t}}} \in [0, 0.5)\) and \(\nu _1({{\hat{t}}}) := 1\), \(\nu _2({{\hat{t}}}) := 5 \pi \cdot (4 ({{\hat{t}}}0.5)  1)\), for \({{\hat{t}}} \in [0.5, 1)\), \({{\hat{t}}} = t  k\), \(k \in {\mathbb {N}}_0\), and, scalars \(a = 50\) and \(s=\frac{1}{3}\). The (analytic) solution (21) mimics a counterclockwise rotating cone which additionally changes its height and orientation over the period \(T=1\). Precisely, the orientation of the cone switches from negative to positive while passing \(t=0.25\) and from positive to negative while passing \(t=0.75\). The inhomogeneous Dirichlet boundary condition, the inhomogeneous initial condition and the righthand side forcing term \({g}\), are calculated from the given analytic solution (21) and Eq. (1), where the latter uses the exact Stokes solution \(\mathbf{v}\) given by Eq. (20). Moreover, we note that the assembly of the transport system matrix uses the approximated fullydiscrete Stokes solution \(\mathbf{v}_{\sigma h}\) of (15) that has to be transferred to the spatial and temporal mesh of the transport problem, cf. Remark 2 in Sect. 5.
Since we study the global spacetime refinement behavior here, we restrict the convection–diffusion–reaction transport problem to a nonstabilized case, i.e. we set \(\delta _0:=0\) within the local SUPG stabilization parameter \(\delta _K = \delta _0 \cdot h_K\), where \(h_K\) denotes the cell diameter of the spatial mesh cell K. Moreover, we set the diffusion coefficient \(\varepsilon =1\) and choose a constant reaction coefficient \(\alpha =1\).
The global spacetime refinement behavior is illustrated by Table 2 and nicely confirms our results with respect to the expected EOCs for the solely solved transport problem obtained in [25, Example 1], cf. columns four and five of Table 2. Furthermore, with regard to the EOCs of the Stokes solution, we note that both approximations cG(2)–dG(0) as well as cG(3)–dG(0) are restricted through the lowest order approximation in time, cf. columns eight and nine of Table 2.
6.2 Example 2 (SpaceTime Adaptivity Studies for the Coupled Problem)
The second example serves to study the goaloriented spacetime adaptivity behavior of our algorithm introduced in Sect. 5. More precisely, the transport problem is adaptively refined in space and time using an approximated Stokes solution \(\mathbf{v}_{\sigma h}\) on a coarser global refined mesh in space and time. The initial spacetime meshes of the transport problem are once more refined compared to the initial meshes of the Stokes flow problem, cf. the first row of Table 3. The temporal and spatial mesh of the Stokes flow problem is refined globally if the global \(L^2(L^2)\)error \(\mathbf{v}\mathbf{v}_{\sigma h}^{2,0}\) is larger than its counterpart \(uu_{\tau h}^{1,0}\) or rather \(uu_{\tau h}^{1,1}\) for the transport problem (cf. columns five and nine of Tables 3 and 4, respectively.).
We study problem (1) and (3) with the given analytical solutions (21) and (20), respectively, with the same settings as given in Sect. 6.1. Our target quantity for the transport problem is chosen to control the global \(L^2(L^2)\)error of e, \(e = {u} {u}_{\tau h}\), in space and time, given by
The tuning parameters of the goaloriented adaptive Algorithm given in Sect. 5 are chosen here in a way to balance automatically the potential misfit of the spatial and temporal errors as
For measuring the accuracy of the error estimator, we will study the socalled effectivity index as the ratio of the estimated error over the exact error, given by
Desirably, the index \({\mathcal {I}}_{\text {eff}}\) should be close to one. In Tables 3 and 4 we present the development of the total discretization error \(J(e)=\Vert e\Vert _{(0,T)\times \varOmega }\) for (22), the approximated spatial and temporal error estimators \(\eta _h\) and \(\eta _{\tau }\) as well as the effectivity index \({\mathcal {I}}_{\mathrm {eff}}\) during an adaptive refinement process for two different primal and dual solution pairings \(\{u_{\tau h},z_{\tau h}\}\): cG(1)–dG(0)/cG(2)–dG(0), cG(1)–dG(1)/cG(2)–dG(1) of the transport problem. Moreover, the development of the total discretization error \(\mathbf{v}\mathbf{v}_{\sigma h}^{2,0}\) for the Stokes flow solution on a global refined mesh in space and time and the corresponding number of slabs and spatial cells is displayed. Thereby, \(\mathbf{v}_{\sigma h}^{2,0}\) corresponds to a Stokes solution approximation in a cG(2)–dG(0) discretization. We use an approximation of the temporal weights by a higherorder reconstruction strategy using GaussLobatto quadrature points. Here and in the following, \(\ell \) denotes the refinement level or DWR loop, N or \(N^{\mathrm {f}}\) the total cells in time, \(N_{K}^{\mathrm {max}}\) or \(N_{K}^{\mathrm {f},\mathrm {max}}\) the number of spatial cells on the finest spatial mesh within the current loop, and \(N_{\mathrm {DoF}}^{\mathrm {tot}}\) or \(N_{\mathrm {DoF}}^{\mathrm {f},\mathrm {tot}}\) the total spacetime degrees of freedom of the transport or Stokes flow problem, respectively.
Regarding the accuracy of the underlying error estimator, as given by the last column of Tables 3 or 4, respectively, we observe a good quantitative estimation of the discretization error as the respective effectivity index increases getting close to one. With regard to efficiency reasons for a spacetime adaptive algorithm, it is essential to ensure an equilibrated reduction of the temporal as well as spatial discretization error, cf. [32, Sec. 3.3]. Referring to this, we point out a good equilibration of the spatial and temporal error indicators \(\eta _{h}\) and \(\eta _{\tau }\) in the course of the refinement process (columns ten and eleven of Tables 3 and 4).
Finally, in Fig. 5 we visualize exemplary the distribution of the adaptively determined time cell lengths \(\tau _K\) of \({\mathcal {T}}_{\tau ,n}\), used for the transport problem, as well as the distribution of the globally determined time cell lengths \(\sigma _K\) of \({\mathcal {T}}_{\sigma ,n}\), used for the Stokes flow problem, over the whole time interval I for different DWR refinement loops, corresponding to Table 3. The initial temporal meshes for the transport and Stokes flow problem are chosen fulfilling the requirements presented in Sect. 2.2 and Fig. 1. While the time steps for the transport problem become smaller when the cone is changing its orientation (\(t=0.25\) and \(t=0.75\)), the time steps for the Stokes flow problem stay comparatively large in the course of the refinement process, cf. the last two plots of Fig. 5. Away from the time points of orientation change, the temporal mesh of the transport problem is almost equally decomposed. This behavior seems natural for a global acting target quantity (22) and nicely confirms our approach of an efficient temporal approximation of a rapidly changing transport coupled with a slowly varying viscous flow.
6.3 Example 3 (Transport in a Channel)
In this example, we simulate a convectiondominated transport with goaloriented adaptivity of a species through a channel with a constraint. The domain and its boundary colorization are presented by Fig. 6. Precisely, the spatial domain is composed of two unit squares and a constraint in the middle which restricts the channel height by a factor of 5. Precisely, \(\varOmega = (1,0)\times (0.5,0.5) \cup (0,1)\times (0.1,0.1) \cup (1,2)\times (0.5,0.5)\) with an initial cell diameter of \(h=\sqrt{2 \cdot 0.025^2}\). The time domain is set to \(I=(0,2.5)\) with an initial \(\tau =0.1\) for the transport and \(\sigma =2.5\) for the Stokes flow problem for the initialization of the slabs for the first loop \(\ell =1\). This choice has been made to compare the results to Example 2 in [25], where a quasistationary Stokes flow solution \(\mathbf{v}_h\) was used. We approximate the primal solution \({u}_{\tau h}^{1,0}\) with the cG(1)–dG(0) method, the dual solution \({z}_{\tau h}^{2,0}\) with the cG(2)–dG(0) method and the Stokes flow solution \(\mathbf{v}_{\sigma h}^{2,0}\) with the cG(2)–dG(0) method.
The target quantity is
The transport of the species, which enters the domain on the left with an inhomogeneous and timedependent Dirichlet boundary condition and leaves the domain on the right through a homogeneous Neumann boundary condition, is driven by the convection with magnitudes between 0 and 5 as displayed in Fig. 7. The diffusion coefficient has the constant and small value of \(\varepsilon =10^{4}\) and the reaction coefficient is chosen \(\alpha =0.1\). The local SUPG stabilization coefficient is here set to \(\delta _K = \delta _0 \cdot h_K\), \(\delta _0=0\), i.e. a vanishing stabilization here. The initial value function \({u}_0=0\) as well as the forcing term \({g}=0\) are homogeneous. The Dirichlet boundary function value is homogeneous on \(\varGamma _D\) except for the line \((1,1) \times (0.4,0.4)\) and time \(0 \le t \le 0.2\) where the constant value
is prescribed on the solution. The viscosity is set to \({\nu }=1\). The tuning parameters of the goaloriented adaptive Algorithm given in Sect. 5 are chosen here in a way to balance automatically the potential misfit of the spatial and temporal errors as \(\theta _h^\mathrm {bottom} = 0\),
The solution profiles and corresponding adaptive meshes of the primal solution \({u}_{\tau h}^{1,0}\) of the loop \(\ell =8\) for \(t=0.15\), \(t=0.70\), \(t=0.92\), \(t=1.33\) and \(t=2.45\) are given by Fig. 8. The refinement in space is adjusted to the position of the transported species within the channel. It is located to the layers of the transported species, whereas the mesh stays coarse in the nonaffected area. In Fig. 9 we present a comparative study of the solution profile and corresponding meshes for \(t=0.95\) over the adaptivity loops. For \(\ell =1,2,3\) obvious spurious oscillations in the left square and at the beginning of the constriction are existing, which are captured and resolved by the goaloriented adaptivity by taking spatial mesh refinements along the layers of the transported species within the left square and within the constriction of the channel. For \(\ell >3\) the spatial refinements capture especially the solution profile fronts with strong gradients with a focus on the highconvective middle of the spatial domain. In Fig. 10 we visualize the temporal distribution of the transport problem for several DWRloops. The time cell lengths of the Stokes flow problem is kept fixed with value \(\sigma _K=2.5\) for all DWRloops here and thus explicitly not displayed. We observe an adaptive refinement in time at the beginning, consistent with the restriction in time of the inflow boundary condition. The closer we get to the final time point T the coarser the temporal mesh is chosen.
The refinement in space and time is automatically balanced due to the dynamic choice of \(\theta _h^\mathrm {top}\) and \(\theta _\tau ^\mathrm {top}\) given by (25) and is illustrated by Table 5. Regarding the spatial and temporal error indicators (cf. columns five and six of Table 5) a good equilibration can be observed within the final loop, whereas in the first step a mismatch occurs resulting in a solely temporal refinement between \(\ell =1\) and \(\ell =2\).
Finally, we modify the parabolic inflow condition for the Stokes flow problem in order to investigate our multirate in time approach for the present example. More precisely, on the left boundary \(\varGamma _{\text {in}}\) the inflow condition \(\mathbf{v}_D\) is now given by
Moreover, for the transport problem, the Dirichlet boundary function value is homogeneous on \(\varGamma _D\) except for the line \((1,1) \times (0.4,0.4)\) and time \(0 \le t \le 0.1\) where the constant value
is prescribed on the solution. Therefore, the time domain \(I=(0,2.5)\) is now discretized with the same initial \(\tau =\sigma =0.1\) for the transport and the Stokes flow problem for the first loop \(\ell =1\). In Fig. 11 we visualize the distribution of the adaptively determined time cell lengths \(\tau _K\) and \(\sigma _K\) used for the transport and Stokes flow problem, respectively, over the whole time interval I for different DWR refinement loops. We observe a similar behavior as displayed in Fig. 10. The temporal mesh is refined close to the time conditions of the respective inflow boundaries for both problems, where the refinement in time for the Stokes flow problem is chosen to refine those slabs related to the inflow condition (26) for each second DWRloop. Away from the temporal inflow condition both temporal meshes stay coarse.
7 Conclusion
In this work, we presented a multirate in time approach regarding different time scales for a rapidly changing transport coupled with a slowly creeping Stokes flow. The transport problem is represented by a convectiondominated convection–diffusion–reaction equation which is for this reason stabilized using the residual based SUPG method. Both subproblems are discretized using a discontinuous Galerkin method dG(r) with an arbitrary polynomial degree \(r \ge 0\) in time and a continuous Galerkin method cG(p) with an arbitrary polynomial degree \(p \ge 1\) in space. A goaloriented a posteriori error representation based on the Dual Weighted Residual method was derived for the transport problem. This error representation is splitted into an amount in space and time whose localized forms serve as error indicators for the adaptive mesh refinement process in space and time. The temporal weights of the DWR adaptivity process are approximated by a higherorder reconstruction approach whereas the spatial weights are approximated by higherorder finite elements. The practical realization of the spacetime slabs is based on tensorproduct spaces which enables for an efficient and flexible software implementation of the underlying approach. In numerical experiments we verified expected experimental orders of convergence of the underlying subproblems as well as the coupled problem. Furthermore, spacetime adaptivity studies for the coupled problem were investigated for an academic test problem as well as a problem of practical interest, leading to highefficient adaptively refined meshes in space and time. Effectivity indices close to one and wellbalanced error indicators in space and time were obtained. Spurious oscillations that typically arise in numerical approximations of convectiondominated problems could be reduced significantly. Although the computation of the estimators causes additional computational costs, the numerical results give a first hint of the potential regarding a temporal mesh that is adapted to the dynamics of the active or fast components compared to a more standard fixedtime strategy where the whole mesh is refined due to these fast components. Moreover, coupled systems with strongly differing time scales might even not become feasible without multirate timestepping approaches. Finally, the here presented approach for coupled free flow and species transport is fairly general and can be easily adopted to other multiphysics systems coupling phenomena that are characterized by strongly differing time scales.
Data Availibility
All data generated or analysed during this study are included in this published article.
References
Jammoul, M., Wheeler, M.F., Wick, T.: A phasefield multirate scheme with stabilized iterative coupling for pressure driven fracture propagation in porous media. Comput. Math. Appl. 91, 176–191 (2021)
Ge, Z., Ma, M.: Multirate iterative scheme based on mutiphysics discontinuous Galerkin method for a poroelasticity model. Appl. Numer. Math. 128, 125–138 (2018)
Gupta, S., Wohlmuth, B., Helmig, R.: Multirate time stepping schemes for hydrogeomechanical model for subsurface methane hydrate reservoirs. Adv. Water Res. 91, 78–87 (2016)
Almani, T., Kumar, K., Dogru, A., Singh, G., Wheeler, M.F.: Convergence analysis of multirate fixedstress split iterative schemes for coupling flow with geomechanics. Comput. Meth. Appl. Mech. Eng. 311, 180–207 (2016)
Gear, C.W., Wells, D.R.: Multirate linear multistep methods. BIT 24(4), 484–502 (1984)
Günther, M., Rentrop, P.: Multirate ROW methods and latency of electric circuits. Appl. Numer. Math. 13(1), 83–102 (1993)
Gander, M.J., Halpern, L.: Techniques for locally adaptive time stepping developed over the last two decades. In: Bank, R., Holst, M., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering XX, Lecture Notes in Computational Science and Engineering, vol. 91, pp. 377–385. Springer, Berlin, (2013)
Köcher, U., Bruchhäuser, M.P., Bause, M.: Efficient and scalable data structures and algorithms for goaloriented adaptivity of spacetime FEM codes. Software X 10, 100239 (2019)
Arndt, D., Bangerth, W., Blais, B., Fehling, M., Gassmöller, R., Heister, T., Heltai, L., Köcher, U., Kronbichler, M., Maier, M., Munch, P., Pelteret, J.P., Proell, S., Simon, K., Turcksin, B., Wells, D., Zhang, J.: The deal.II Library, Version 9.3. J. Numer. Math. 29(3), 171–186 (2021)
Larson, M.G., Målquist, A.: Goal oriented adaptivity for coupled flow and transport with applications in oil reservoir simulations. Comput. Methods Appl. Mech. Eng. 196, 3546–3561 (2007)
Allaire, G.: Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2, 203–222 (1989)
Wick, T.: Goal functional evaluations for phasefield fracture using PUbased DWR mesh adaptivity. Comput. Mech. 57, 1017–1035 (2016)
Odsæter, L.H., Kvamsdal, T., Larson, M.G.: A simple embedded discrete fracturematrix model for a coupled flow and transport problem in porous media. Comp. Methods Appl. Mech. Eng. 343, 572–601 (2019)
Bengzon, F., Larson, M.G.: Adaptive finite element approximation of multiphysics problems: a fluidstructure interaction model problem. Int. J. Numer. Methods Eng. 84, 1451–1465 (2010)
Gujer, W.: Systems Analysis for Water Technology. Springer, Berlin (2008)
Morgenroth, E.: How are characteristic times (\(\tau _{\it char\it }\)) and nondimensional numbers related. https://ethz.ch/content/dam/ethz/specialinterest/baug/ifu/watermanagementdam/documents/education/Lectures/UWM3/SAMM.HS15.Handout.CharacteristicTimes.pdf (2015). Accessed 07 February 2022
Burman, E.: Robust error estimates in weak norms for advection dominated transport problems with rough data. Math. Models Methods Appl. Sci. 24(13), 2663–2684 (2014)
John, V., Knobloch, P., Novo, J.: Finite elements for scalar convectiondominated equations and incompressible flow problems: A never ending story? Comput. Vis. Sci. 19, 47–63 (2018)
John, V., Novo, J.: Error analysis of the SUPG finite element discretization of evolutionary convectiondiffusionreaction equations. SIAM J. Numer. Anal. 49(3), 1149–1176 (2011)
John, V., Schmeyer, E.: Finite element methods for timedependent convectiondiffusionreaction equations with small diffusion. Comput. Methods Appl. Mech. Eng. 198, 173–181 (2009)
Becker, R., Rannacher, R.: An optimal control approach to a posteriori error estimation in finite element methods. In: Iserles, A. (ed.) Acta Numerica, vol. 10, pp. 1–102. Cambridge University Press, Cambridge (2001)
Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Basel (2003)
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, New York (2000)
Verfürth, R.: A Review of A Posteriori Error Estimation and Adaptive MeshRefinement Techniques. WileyTeubner Series Advances in Numerical Mathematics. WileyTeubner, New York (1996)
Bause, M., Bruchhäuser, M.P., Köcher, U.: Flexible goaloriented adaptivity for higherorder spacetime discretizations of transport problems with coupled flow. Comput. Math. Appl. 91, 17–35 (2021)
John, V.: Finite Element Methods for Incompressible Flow Problems. Springer Series in Computational Mathematics, vol. 51. Springer, Cham (2016)
Roos, H.G., Stynes, M., Tobiska, L.: Robust Numerical Methods for Singularly Perturbed Differential Equations. Springer, Berlin (2008)
Ern, A., Guermond, J.L.: Finite Elements III: FirstOrder and Timedependent PDES. Texts in Applied Mathematics, vol. 74. Springer, Cham (2021)
Carey, G.F., Oden, J.T.: Finite Elements, Computational Aspects, Vol. III (The Texas finite element series). PrenticeHall, Englewood Cliffs, New Jersey (1984)
Hughes, T.J.R., Brooks, A.N.: A multidimensional upwind scheme with no crosswind diffusion. In: Hughes, T.J.R. (Ed.) Finite Element Methods for Convection Dominated Flows, AMD, Vol. 34, pp. 19–35. The American Society of Mechanical Engineers (ASME) (1979)
Brooks, A.N., Hughes, T.J.R.: Streamline upwind/PetrovGalerkin formulations for convection dominated flows with particular emphasis on the incompressible NavierStokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)
Besier, M., Rannacher, R.: Goaloriented spacetime adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. Int. J. Numer. Methods Fluids 70(9), 1139–1166 (2012)
Bruchhäuser, M.P., Schwegler, K., Bause, M.: Dual weighted residual based error control for nonstationary convectiondominated equations: Potential or ballast? In: Barrenechea, G.R., Mackenzie, J. (eds.) Boundary and Interior Layers, Computational and Asymptotic Methods BAIL 2018. Lecture Notes in Computational Science and Engineering, vol. 135, pp. 1–17. Springer, Cham (2020)
Brenner, A., Bänsch, E., Bause, M.: Apriori error analysis for finite element approximations of Stokes problem on dynamic meshes. IMA J. Numer. Anal. 34, 123–146 (2014)
Acknowledgements
The authors wish to thank the anonymous reviewers for their help to improve the presentation of this paper.
Funding
Open Access funding enabled and organized by Projekt DEAL. The authors have not disclosed any funding.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
The authors have no relevant financial or nonfinancial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 Galerkin Orthogonality for Temporal and Spatial Error of Transport Problem
For the temporal error \(e = {u}{u}_{\tau }\) we get the following Galerkin orthogonality by subtracting Eq. (8) from Eq. (5)
with a nonvanishing righthand side term depending on the the temporal error in the approximation of the flow field.
For the spatial error \(e = {u}_{\tau }{u}_{\tau h}\) we get the following Galerkin orthogonality by subtracting Eq. (14) from Eq. (8)
with a nonvanishing righthand side term depending on the stabilization and the spatial error in the approximation of the flow field.
1.2 Lagrangian Functionals and Residuals
The Lagrangian functionals \({\mathcal {L}}: X\times X \rightarrow {\mathbb {R}}\), \({\mathcal {L}}_\tau : X_{\tau }^{r} \times X_{\tau }^{r} \rightarrow {\mathbb {R}}\), and \({\mathcal {L}}_{\tau h}: X_{\tau h}^{r,p} \times X_{\tau h}^{r,p} \rightarrow {\mathbb {R}}\) are defined by
Here, the Lagrange multipliers \({z}\), \({z}_\tau ,\) and \({z}_{\tau h}\) are called dual variables in contrast to the primal variables \({u}\), \({u}_\tau ,\) and \({u}_{\tau h}\); cf. [21, 32].
Considering the directional derivatives of the Lagrangian functionals, also known as Gâteaux derivatives, with respect to their first argument, i.e.
leads to the socalled dual problems: Find the continuous dual solution \({z}\in X\), the semidiscrete dual solution \({z}_{\tau } \in X_{\tau }^{r}\) and the fully discrete dual solution \({z}_{\tau h} \in X_{\tau h}^{r,p}\), respectively, such that
where we refer to our work [25] for a detailed description of the adjoint bilinear forms \(A^{\prime },A_{\tau }^{\prime },A_S^{\prime }\) as well as the dual right hand side term \(J^\prime \).
The primal and dual residuals based on the continuous and semidiscrete schemes are defined by means of the Gâteaux derivatives of the Lagrangian functionals in the following way:
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bruchhäuser, M.P., Köcher, U. & Bause, M. On the Implementation of an Adaptive Multirate Framework for Coupled Transport and Flow. J Sci Comput 93, 59 (2022). https://doi.org/10.1007/s1091502202026z
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s1091502202026z