Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 938727 |

Viviane Klein, Malgorzata Peszynska, "Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model", Journal of Applied Mathematics, vol. 2012, Article ID 938727, 26 pages, 2012.

Adaptive Double-Diffusion Model and Comparison to a Highly Heterogeneous Micro-Model

Academic Editor: Khalida Inayat Noor
Received21 Mar 2012
Accepted13 Apr 2012
Published08 Aug 2012


Double-diffusion model is used to simulate slightly compressible fluid flow in periodic porous media as a macro-model in place of the original highly heterogeneous micro-model. In this paper, we formulate an adaptive two-grid numerical finite element discretization of the double-diffusion system and perform a comparison between the micro- and macro-model. Our numerical results show that the micro-model solutions appear to converge to the macro-model linearly with the parameter ε of periodic geometry. For the two-grid discretization, the a priori and a posteriori error estimates are proved, and we show how to adapt the grid for each component independently.

1. Introduction

When modeling phenomena in highly heterogeneous media, one frequently finds that the coefficients of differential equations describing these phenomena vary by several orders of magnitude between close-by locations. Consequently, the solutions to the exact, that is, micro-models, vary at multiple separate spatial and time scales, and it is convenient to work with their spatial and temporal averages, that is, with the solutions to macro-models. Various multiscale modeling techniques have been introduced with the aim to derive, analyze, and approximate micro-models by the macro-models. In particular, the mathematical framework for construction and analysis of the average, that is, homogenized, models for multiscale media is well understood, see, for example, [13]. However, there are few results devoted to the comparison between the micro- and macro-models for evolution equations and to their adaptive numerical discretizations.

Consider the following transient diffusion model which describes the density 𝜌 of single-phase slightly compressible flow in a porous domain Ω: 𝜙𝜕𝜌𝜕𝑡(𝑘𝜌)=𝑓,inΩ,𝑡>0,(1.1) where the coefficients 𝜙, 𝑘 denote, respectively, the porosity and conductivity.

In this paper, we propose a two-grid adaptive finite element approximation of the following multiscale averaged model for (1.1) in highly heterogeneous media such as porous media with fractures𝜙1𝜕̃𝜌1̃𝑘𝜕𝑡1̃𝜌1+𝑐̃𝜌1̃𝜌2=𝜙𝑓,inΩ,𝑡>0,(1.2a)2𝜕̃𝜌2̃𝑘𝜕𝑡2̃𝜌2+𝑐̃𝜌2̃𝜌1=0,inΩ,𝑡>0,(1.2b)proposed in [4, 5]. The system (1.2a) and (1.2b) describes the average behavior of fluids in media with two or more subregions of distinct features, where the subregions may or may not be connected globally at micro-scale. At macro-scale the interaction of fluids associated with these subregions is modeled with the exchange term 𝑐(̃𝜌2̃𝜌1), and the coefficients ̃𝑘𝑖, 𝜙𝑖 are computed from micro-scale geometry and coefficients. The system (1.2a) and (1.2b) is useful in applications in porous media as well as in description of other multiscale evolution phenomena in heterogeneous media, for example, heat conduction. In addition, its special, degenerate, case of (1.2a) and (1.2b) with ̃𝑘2=0 known as the Warren-Root model [6]𝜙1𝜕̃𝜌1̃𝑘𝜕𝑡1̃𝜌1+𝑐̃𝜌1̃𝜌2=𝜙𝑓,inΩ,𝑡>0,(1.3a)2𝜕̃𝜌2𝜕𝑡+𝑐̃𝜌2̃𝜌1=0,inΩ,𝑡>0,(1.3b)is a popular way of modeling of subscale diffusion accompanying advection and adsorption, see [710].

The macro-model (1.2a) and (1.2b) of interest to this paper was introduced in [4] based on heuristic arguments. The model, while intuitively clear and useful for practical purposes, for a long time was lacking a rigorous multiscale analysis and interpretation via an associated appropriate micro-model. Only recently in [11] it was shown rigorously by two-scale convergence that (1.2a) and (1.2b) is indeed a homogenized limit of (1.1) in certain highly heterogeneous media. Most interesting is that the original microstructure which leads to (1.2a) and (1.2b) is composed of three rather than two media as in [12]. Also, the derivation in [11] demonstrates clearly that in two-dimensional geometries and more generally for quasistatic models in disconnected microscale subregions we have ̃𝑘2=0.

However, the analysis in [11] does not include a direct comparison of (1.1) and (1.2a) and (1.2b), and this motivated our research. We compare the computational solutions to (1.2a) and (1.2b) and (1.1) quantitatively in function of the microstructure parameters. In addition, since typically ̃𝑘02̃𝑘1, it is natural to suggest that in numerical simulation ̃𝜌1 and ̃𝜌2 should be approximated on separate grids, since they evolve differently and since this can lead to more efficient numerical algorithms. In fact, the spatial grids should be chosen adaptively guided by some a posteriori estimators and should be allowed to vary in time. The idea of using two grids extends that in [13] where we considered a stationary analogue of (1.2a) and (1.2b) and proved upper and lower bounds for residual type a posteriori estimators. The estimators proposed in [13] were shown to be robust, that is, insensitive to ̃𝑘20 and to other parameters. In this paper we extend those to the evolution system (1.2a) and (1.2b) and present an a posteriori error estimator of residual type based partly on the work in [1416] for scalar equations.

The aim of this paper is thus twofold, and the paper is organized as follows. In Section 2 we introduce the notation and the generic flow model as well as its numerical approximation. In Section 3 we develop the details of micro- and macro- models and compare their discrete solutions in function of a characteristic parameter of the microstructure 𝜀. In Section 4, we define the numerical approximations to these models and formulate the a-priori and a-posteriori estimates for the approximation error. We then illustrate the use of a-posteriori estimators with an adaptive example. The appendix provides the technical proofs of some of the results derived in this paper.

2. Notation and Preliminaries on the Flow Model

Here we introduce the notation and preliminaries. We follow, for example, [17, 18]. Let the domain of flow Ω𝑑, 𝑑=2,3, be a bounded polygonal region with piecewise smooth boundary 𝜕Ω=Γ𝐷Γ𝑁, where Γ𝐷, Γ𝑁 are disjoint regions of the boundary 𝜕Ω where Dirichlet and Neumann boundary conditions are prescribed, respectively.

For any subset 𝜔Ω, we use the standard notation for Lebesgue 𝐿2(𝜔) and Sobolev spaces 𝐻𝑘(𝜔), 𝑘. These are equipped with the usual seminorms ||𝑘,𝜔, norms 𝑘,𝜔=𝐻𝑘(𝜔) and the usual scalar product (s) (𝑓,𝑔)𝜔=(𝑓,𝑔)𝐿2(𝜔). We denote 𝑓𝜔=(𝑓,𝑓)𝜔1/2. If 𝜔=Ω, the subscript 𝜔 will be omitted. We also let 𝐻1𝐷(Ω)={𝑣𝐻1(Ω)𝑣|Γ𝐷=0} under the standard norm in 𝐻1(Ω), 𝑣1=(𝑣2+𝑣2)1/2.

For any 𝑇>0 and any vector space 𝑆, the space 𝐿2(0,𝑇;𝑆) consists of all square-integrable functions with values in 𝑆 such that 𝑣𝐿2(0,𝑇;𝑆)=(𝑇0𝑣2𝑆)1/2 is finite. The space 𝐶([0,𝑇];𝑆) is defined similarly.

2.1. Variational Form of Micro-Model

Consider (1.1) as a model for single-phase slightly compressible flow in porous media in which the gravity terms have been ignored; see [19] for derivation. Here 𝜙=𝜙(𝑥) and 𝑘=𝑘(𝑥),𝑥Ω denote the nonnegative coefficients of porosity and conductivities, respectively, and 𝑓 accounts for external sources. We assume for simplicity that 𝑘 is a scalar quantity.

It is well known that (1.1) is well posed if appropriate boundary and initial data are imposed. We will use 𝜌=𝜌0,inΩ×{0},𝜌=𝜌𝐷,onΓ𝐷,𝑘𝜌𝑛=0,onΓ𝑁.(2.1) For simplicity we assume that 𝜌𝐷0 and that the flow is driven by 𝑓. However, in simulation examples we use 𝑓0, 𝜌𝐷0.

The variational formulation of problem (1.1) and (2.1) reads

Find 𝜌𝐶([0,𝑇];𝐿2(Ω))𝐿2(0,𝑇;𝐻1𝐷(Ω))  so that𝜙𝜕𝜌𝜕𝑡,𝜓+(𝑘𝜌,𝜓)=(𝑓,𝜓),𝜓𝐻1𝐷(Ω).(2.2) For any 𝜌0𝐿2(Ω), 𝑓𝐶([0,𝑇];𝐿2(Ω)), there is a unique solution 𝜌𝐶([0,𝑇];𝐿2(Ω))𝐿2(0,𝑇;𝐻1𝐷(Ω)) to (2.7); see [20, Theorem  11.1.1, page 366]).

2.2. Variational Form of Macro-Model

Now consider (1.2a) and (1.2b) and assume that 𝜙1, 𝜙2, ̃𝑘1, ̃𝑘2, and 𝑐𝐿(Ω) are bounded uniformly from above and below by positive constants and are independent of time. We consider separately the particular case of (1.3a) and (1.3b) when ̃𝑘20.

Our variational formulation for (1.2a) and (1.2b) uses the theory developed in [17, 18] for evolution problems with m-accretive operators by the Hille-Yosida Theorem. For analysis of classical solutions see [21, 22].

For the macro-model (1.2a) and (1.2b) and if ̃𝑘2>0, we assume the following initial and boundary conditions defined for 𝑖=1,2: ̃𝜌𝑖=̃𝜌𝑖,0,inΩ×{0},̃𝜌𝑖=𝜌𝐷,onΓ𝐷,̃𝑘𝑖̃𝜌𝑖𝑛=0,onΓ𝑁.(2.3) As before, we assume that 𝜌𝐷0 for now.

Let ̃𝜌𝑖(,0)=̃𝜌𝑖,0𝐿2(Ω), 𝑓𝐶([0,𝑇],𝐿2(Ω)), 𝑢𝐷0. The variational form of (1.2a), (1.2b), and (2.3) is as follows:

Find ̃𝜌𝑖𝐶([0,𝑇];𝐿2(Ω))𝐿2(0,𝑇;𝐻1𝐷(Ω)),  𝑖=1,2, such that𝜙1𝜕̃𝜌1+̃𝑘𝜕𝑡,𝜓1̃𝜌1+𝑐,𝜓̃𝜌1̃𝜌2,𝜓=(𝑓,𝜓),𝜓𝐻1𝐷𝜙(Ω),2𝜕̃𝜌2+̃𝑘𝜕𝑡,𝜉2̃𝜌2+𝑐,𝜉̃𝜌2̃𝜌1,𝜉=0,𝜉𝐻1𝐷(Ω).(2.4) The well posedness of this system (2.4) follows immediately by showing that the operator 𝒟𝐻1𝐷(Ω)×𝐻1𝐷(Ω)𝐿2(Ω)×𝐿2(Ω)̃𝑘=𝜕1̃𝑘𝜕+𝑐𝐼𝑐𝐼𝑐𝐼𝜕2𝜕+𝑐𝐼,where𝐼istheidentityoperator,(2.5) is m-accretive. The accretiveness is straightforward by showing that (𝐵𝑢,𝑢)0 in the product space. The maximal property follows from that of the operators ̃𝑘𝜕𝑖𝜕, 𝐼.

Now, if ̃𝑘20, we do not impose boundary conditions on ̃𝜌2. The variational formulation of (3) and (2.3), reads

Find ̃𝜌1𝐶([0,𝑇];𝐿2(Ω))𝐿2(0,𝑇;𝐻1𝐷(Ω)),  ̃𝜌2𝐶([0,𝑇];𝐿2(Ω))𝐿2(0,𝑇;𝐿2(Ω))   such that𝜙1𝜕̃𝜌1+̃𝑘𝜕𝑡,𝜓1̃𝜌1+𝑐,𝜓̃𝜌1̃𝜌2,𝜓=(𝑓,𝜓),𝜓𝐻1𝐷𝜙(Ω),2𝜕̃𝜌2+𝑐𝜕𝑡,𝜉̃𝜌2̃𝜌1,𝜉=0,𝜉𝐿2(Ω).(2.6) The well posedness of (2.6) follows similarly from the Hille-Yosida theorem. Also, ̃𝜌2 has the minimum of regularity of its initial data and of ̃𝜌1.

2.3. Notation on Numerical Discretization

For computations of solutions we use conforming (Galerkin) finite elements for spatial discretization and implicit Euler time stepping. We partition the time interval [0,𝑇] into subintervals [𝑡𝑛1,𝑡𝑛], 1𝑛𝑁, such that 0=𝑡0<𝑡1<<𝑡𝑁=𝑇. We denote Δ𝑡𝑛=𝑡𝑛𝑡𝑛1 and 𝑧(𝑡𝑛)=𝑧𝑛 for any function 𝑧.

For spatial discretization, we adopt standard finite element nomenclature that can be found in textbooks such as [2325]. At each time step 𝑛, we denote by 𝒯𝑛, >0, a family of admissible and shape-regular partitions of Ω into a finite number of elements. For any element 𝑇𝒯𝑛, we let 𝑇 be the diameter of 𝑇 and denote =max𝑇𝒯𝑇. Also, 𝑛 is the set of all edges of 𝒯𝑛. For any edge 𝐸𝑛, we let 𝐸 denote the diameter of the edge 𝐸. Finally, we denote by 𝒫𝑘(𝑇) the space of polynomials of degree 𝑘 in 𝑑.

Numerical Approximation of Micro-Model
At each time step 𝑛, we define 𝑉𝑛=𝑣𝐶Ω𝑇𝒯𝑛,𝑣𝑇𝒫𝑘(𝑇),𝑣Γ𝐷=0,(2.7) and seek an approximation 𝜌𝑛𝑉𝑛 such that for all 𝜓𝑉𝑛𝜙𝜌𝑛𝜌𝑛1Δ𝑡𝑛,𝜓+𝑘𝜌𝑛,𝜓=𝑓,𝜓𝜌,(2.8a)0=𝜌0inΩ.(2.8b)Here denotes an interpolation or projection operator into 𝑉𝑛.
It is standard [20] that there is a unique solution 𝜌𝑛𝑉𝑛 for (2.8a) at each 𝑛, 1𝑛𝑁. A priori estimates for the error between the solution to (2.2) and that of (2.8a) and (2.8b) can be found in the literature, see, for example, [20, 25, 26]. For a posteriori error estimates for parabolic problems see [15, 2729]; the methods in the latter two papers are most relevant to our paper.

Numerical Approximation of Macro-Model
Consider now a fully discrete approximation to the solutions to (2.4) or (2.6). The formulation with a fixed grid immediately extends (2.8a) and (2.8b), and appropriate error estimates can be proven. More generally, each of ̃𝜌𝑖 can be associated with its own grid 𝒯𝑛𝑖; see details in Section 4.

3. Comparison of Micro- and Macro-Models

The difficulties associated with numerical approximation (2.8a) and (2.8b) to (1.1) arise when 𝜙, 𝑘 are highly varying coefficients. In particular, consider a porous domain Ω composed of three disjoint regions Ω𝑖, 𝑖=1,,3, see Figure 1, with positive constants (𝜙𝑖,𝑘𝑖)3𝑖=1 so that 𝜙(𝑥)=𝜙1𝜒1(𝑥)+𝜙2𝜒2(𝑥)+𝜙3𝜒3𝑘(𝑥),(𝑥)=𝑘1𝜒1(𝑥)+𝑘2𝜒2(𝑥)+𝑘3𝜒3(𝑥).(3.1) Here 𝜒𝑖 denotes the characteristic function of Ω𝑖.

An accurate numerical approximation to (1.1) with (3.1) requires that the grid 𝒯 is very fine so that it lines up with the interfaces 𝜕Ω𝑖𝜕Ω𝑘. The computational complexity of the associated numerical implementation is very large whenever the geometry of interfaces is complex, but this can be overcome by a modeling approximation. In [11] it is shown that (1.2a) and (1.2b) provides solutions close to those of (1.1), with the coefficients 𝜙𝑖,̃𝑘𝑖 which do not vary locally. Thus a grid for (1.2a) and (1.2b) can be chosen depending on the global macrodynamics rather than local microdynamics and, consequently, it can be much coarser than 𝒯.

3.1. Derivation of Macro-Model

One successful avenue to reduce the complexity of a highly heterogeneous model is to consider a homogenized or upscaled problem, whose coefficients are derived from the original coefficients [1, 2, 30], and which can be simulated on a coarse grid quite accurately. However, for time-dependent problems, this leads to inaccurate representation of the global dynamics, especially if 𝑘1𝑘2 and/or 𝑘1𝑘3.

The double-diffusion models proposed in [4], the relaxation model from [6], and double-porosity models derived in [12] are families of successful modeling strategies for parabolic PDEs with periodic highly heterogeneous media. The recent general derivation in [11] is most general and includes those from [4, 6, 12] as special cases.

The double-diffusion model developed in [4] was shown in [11] to be a limit of a certain micro-model with three regions arranged periodically. The rigorous derivation in [11] allows for each of the regions Ω𝑖 to be connected. Furthermore, it assumes that Ω3 separates Ω1 and Ω2, that is, that 𝜕Ω1𝜕Ω2=. The model in [11] develops a general nonlocal exchange term acting across Ω3. Its special case equivalent to the model in [4] assumes 𝜙30 thereby making a quasistatic assumption and allows to compute the coefficient 𝑐 explicitly.

Double-porosity models proposed in [12] assume that only one of the regions Ω1 is connected and so effectively one can set Ω2=. However, 𝜙3>0. In the macro-model the second equation (1.2b) is replaced by a local micro-model equation, the exchange follows through macro-micro boundary conditions and is nonlocal in nature. We refer to [3135] for various analyses, models, approximations, and extensions of double-porosity models.

Of course, all three regions Ω1, Ω2, and Ω3 can be connected only in 𝑑=3. In 𝑑=2 only one of these regions, say Ω1, can be connected; see Figure 1(a). As we point out below, this implies ̃𝑘2=0, and the macro-model (1.2a) and (1.2b) in 𝑑=2 becomes the Warren-Root relaxation model (3) as in [6].

In this paper we are interested in the double-diffusion models with a constant 𝑐 from [4, 6] and its corresponding quasistatic micro-model from [11]. The exchange region Ω3 is assumed to be small and acts like a thick skin separating Ω1 from Ω2. In consequence, the fluid flows effectively only in two regions Ω1 and Ω2; this is represented by the two sub-equations in (1.2a) and (1.2b). Details, comparison, and simulations are shown in what follows.

3.1.1. Macro-Model Parameters

To compute parameters 𝜙1, 𝜙2, ̃𝑘1, ̃𝑘2, and 𝑐 of the macro-model (1.2a) and (1.2b), one has to calculate local averages and compute auxiliary solutions of differential equations. These depend on 𝜙𝑖, 𝑘𝑖 and on the geometry of Ω𝜀. We follow closely [11].

First we formalize the notion of periodicity and heterogeneity in Ω. Without loss of generality we assume that Ω𝜀1 is globally connected and 𝜙3=0; all other coefficients in (3.1) are positive constants. We follow the usual structure [1, 11, 12]. Let the unit cube 𝑌=(0,1)𝑑𝑑 be divided into three distinct regions 𝑌1, 𝑌2, and 𝑌3, such that 𝜕𝑌1𝜕𝑌2= and 𝑌=3𝑖=1𝑌𝑖 as illustrated by Figure 1(b). Denote by 𝐶𝑖 the characteristic function of 𝑌𝑖, 𝑖=1,2,3, extended 𝑌-periodically to all 𝑑. We assume that the domains {𝑥𝑑𝐶𝑖(𝑥)=1}, 𝑖=1,2,3, have a smooth boundary. Also, we define the 𝜀-periodic characteristic functions 𝐶𝜀𝑖(𝑥)=𝐶𝑖(𝑥/𝜀), 𝑖=1,2,3,  𝑥Ω. Thus, for a fixed 𝜀, Ω is subdivided into three distinct regions Ω𝜀𝑖={𝑥Ω𝐶𝜀𝑖(𝑥)=1}, 𝑖=1,2,3. Another subdivision of Ω is into 𝑂(𝜀𝑑)-periodic cells.

Macroporosities 𝜙𝑖
From [11, equation 16, page 206] we have 𝜙𝑖=𝑌𝑖𝜙𝑖(𝑦)𝑑𝑦.

Macroconductivities ̃𝑘𝑖
From [11, equations 16, 14, page 205-206] we have ̃𝑘𝑝(𝑖,𝑗)=𝑌𝑝𝑘𝑝(𝑦)𝑒𝑖+𝑊𝑝𝑖(𝑦)𝑒𝑗+𝑊𝑝𝑗(𝑦)𝑑𝑦,𝑝=1,2;𝑖,𝑗=1,𝑑,(3.2) where 𝑊𝑝𝑖, for 𝑝=1,2;𝑖=1,,𝑑, is the solution to an auxiliary PDE𝑘𝑝𝑒𝑖+𝑊𝑝𝑖(𝑦)=0,in𝑌𝑝𝑘,(3.3a)𝑝𝑒𝑖+𝑊𝑝𝑖(𝑦)𝑛𝑝=0,onΓ𝑝,3𝑊,(3.3b)𝑝𝑖,𝑘𝑝𝑊𝑝𝑖𝑛𝑝areperiodic.(3.3c)One can prove [1, 3, 11] that ̃𝑘𝑝 is a symmetric matrix.
Now consider 𝑑=2. Here the region Ω𝜀2 cannot be connected. Thus, the boundary condition (3.3c) does not apply and (3.3a), (3.3b), and (3.3c) admit the trivial solution 𝑒𝑖+𝑊2𝑖(𝑦)0 for 𝑖=1,2. Consequently, ̃𝑘2 is the 2×2 null matrix, and the model (1.2a) and (1.2b) becomes the Warren-Root model (3) proposed in [6].

Exchange Term Parameter 𝑐
In general, the flow between Ω1 and Ω2 across Ω3 has transient character, and an appropriate term describing it must be nonlocal in nature.
However, for very small 𝜙3, the flow is of quasistatic nature as discussed in [4]. The exchange term then has the form 𝑐(̃𝜌1̃𝜌2). To compute 𝑐, we consider the solution 𝑈 of𝑘3𝑈=0,in𝑌3,(3.4a)𝑈=1,onΓ1,3,(3.4b)𝑈=0,onΓ2,3,(3.4c)𝑈,𝑘3𝑈𝑛3areperiodiconΓ3,3.(3.4d)The parameter 𝑐 see [11, equation 19, page 209] is given by 𝑐=Γ1,3𝑘3𝑦𝑈𝑛3𝑑𝑠.(3.5) We note that the general non-quasistatic case is when 𝜙3(𝜕𝑈/𝜕𝑡) is included in (3.4d) thereby changing the character of exchange term to nonlocal. Conversely, (3.4a), (3.4b), (3.4c), and (3.4d) can be considered as its special case when 𝜙30.

3.1.2. Numerical Computation of Macro-Model Parameters

The solutions of the auxiliary PDEs (3.3a), (3.3b), and (3.3c) and (3.4a), (3.4b), (3.4c), and (3.4d) are approximated using finite elements with 𝑘11=40 and 𝑐1=20. See [36] for treatment of periodic boundary conditions.

In what follows we ignore the distinction between the exact and numerical values of macro-model parameters ̃𝑘𝑖, 𝑐 and provide examples of their calculations for various values of 𝑘3 and choices of geometry of 𝑌.

In all examples 𝑑=2 hence ̃𝑘2=0000. Also, in all examples 𝑘1const and the geometry of 𝑌 is axisymmetric, and thus ̃𝑘1 is a scalar constant.

Example 3.1. Here 𝑌2=(0.3,0.7)2,  𝑌3=(0.2,0.8)2𝑌2,  𝜙=1𝜒1+104𝜒2,  and 𝑘=1𝜒1+104𝜒2+107𝜒3. We obtain 𝜙1𝜙=0.64,2=1.6×105,̃𝑘1=0.4519,𝑐=1.8634×106.(3.6)

Example 3.2. Let 𝑌2=(0.3,0.7)2,  𝑌3=(0.2,0.8)2𝑌2. 𝜙=1𝜒1+104𝜒2, and  𝑘=1𝜒1+101𝜒2+104𝜒3. We obtain 𝜙1𝜙=0.64,2=1.6×105,̃𝑘1=0.4519,𝑐=1.8634×103.(3.7)

Example 3.3. Here 𝑌2=(0.3,0.7)2,  𝑌3=(0.2,0.8)2𝑌2,  𝜙=1𝜒1+104𝜒2,  and 𝑘=1𝜒1+101𝜒2+102𝜒3. We obtain 𝜙1𝜙=0.64,2=1.6×105,̃𝑘1=0.4519,𝑐=1.8634×101.(3.8)

Example 3.4. Let 𝑌2=(0.3,0.7)2,  𝑌3=(0.1,0.9)2𝑌2,  𝜙=1𝜒1+104𝜒2, and 𝑘=1𝜒1+101𝜒2+104𝜒3. We get 𝜙1𝜙=0.36,2=1.6×105,̃𝑘1=0.2130,𝑐=1.0412×103.(3.9)

Example 3.5. 𝑌2=(0.25,0.75)2,  𝑌3=(0.2,0.8)2𝑌2,  𝜙=1𝜒1+104𝜒2, and 𝑘=1𝜒1+101𝜒2+104𝜒3. We get 𝜙1𝜙=0.64,2=2.5×105,̃𝑘1=0.4519,𝑐=4.3000×103.(3.10)

3.2. Comparison of the Models

Now we can compare the solutions of the micro-model (1.1) to those of the macro-model (1.2a) and (1.2b). In [11] it was shown that the latter two-scale converge to the former as 𝜀0. In addition, the analysis in [11] suggests that it is appropriate to compare ̃𝜌𝑖 to 𝜒𝑖𝜌𝜀. However, this notion of convergence does not give information about the rate at which ̃𝜌𝑖𝜒𝑖𝜌𝜀 may converge, and it involves special periodic test functions.

In this paper we estimate this rate by comparing their numerical approximations directly without the use of any test functions but rather in a certain metric of interest. To the best of our knowledge such comparison or convergence rate was not discussed elsewhere.

Strictly speaking, the two-scale convergence proof considered in [11], and a similar proof for the double-porosity model in [12], includes scaling of 𝑘3 with 𝜀2. This scaling is a formal device necessary to preserve certain parts of the boundary value problem under investigation in the limit as 𝜀0. However, the homogenization limit is intended to serve only as an approximation of the true model, which has a given fixed set of parameters. We do not include this scaling in our computations. Rather, we treat each example, for a given 𝜀, as a data set in its own merit, rather than as an element of a sequence intended to two-scale converge to the limit.

3.2.1. Setup of Computational Experiments

We set up simulations for comparison using compatible data for micro- and macro-models; we set 𝑓0 and choose initial and boundary data driving the flow with interesting dynamics. For porosities and conductivities, we use the values computed in Section 3.1.2.

Let Ω=(0,1)2 and Γ𝐷={0}×[0,1]{1}×[0,1]. On Γ𝐷 we define 𝜌𝐷(0,𝑦)=1,𝜌𝐷(1,𝑦)=0,𝑦[0,1]. On the lateral sides of Ω the Neumann no-flow condition is imposed. Also, let 𝜌01. Thus the flow in the micro-model goes from left to right, and the solution evolves towards the stationary solution 1𝑥,1𝑥. Due to the incompatibility between initial and boundary data, we have high gradients of the solution close to 𝑥=1 for small 𝑡.

We fix the final time 𝑇=0.05 and use uniform time stepping with Δ𝑡=104. The solution of the micro-model 𝜌 depends on the number of cells in ΩΩ𝜀, with 𝜀(0,1]. For example, if 𝜀=1/2, the domain Ω is composed of 2×2=4 cells. We denote the numerical solution of (2.8a) and (2.8b) at 𝑡𝑛 by 𝜌𝑛𝜀, where denotes the grid parameter for the micro-model.

For the macro-model, we use ̃𝜌𝑖,0=𝜌01. Also, we use ̃𝜌𝐷,𝑖=𝜌𝐷 for 𝑖=1,2 when ̃𝑘20. If ̃𝑘20, then the boundary condition for ̃𝜌2 is not prescribed.

3.2.2. Qualitative Comparison

In our first comparison we use parameters from Example 3.1. We solve the macro-model (4.2) for {̃𝜌𝑛1,1,̃𝜌𝑛2,2}𝑁𝑛=1 and the micro-model (2.8a) and (2.8b) for 𝜌𝑛𝜀,. The plots of 𝜌𝑛 and ̃𝜌𝑛𝑖, are in Figure 2, and a different view is shown in Figure 1(d).

The heterogeneous structure of Ω𝜀 is well visible from the behavior of the micro-model solution 𝜌𝑛. Large gradients of solution are visible on cell boundaries due to the large difference between 𝑘1, 𝑘2, and 𝑘3.

However, the behavior in the fast region is well approximated by ̃𝜌𝑛1,1 which envelopes 𝜌𝑛𝜒1, while ̃𝜌𝑛2,2 envelopes 𝜌𝑛𝜒2 well. This is very well seen in a side view in Figure 2(d).

Quite interesting is the behavior of 𝜌2|𝑥=1. Since in our examples ̃𝑘20, no boundary condition at 𝑥=1 is imposed on 𝜌2. Thus, for small 𝑡, 𝜌2(1,𝑡) evolves from its initial constant value of 1 according to (1.3b) with the input from ̃𝜌1. The latter satisfies, however, the homogeneous boundary condition at 𝑥=1. In particular, with constant 𝜙2, 𝑐, we have ̃𝜌2(𝑥,𝑦,𝑡)=̃𝜌2,0(𝑥,𝑦)𝑒(𝑐/𝜙2)𝑡+𝑡0̃𝜌1(𝑥,𝑦,𝑠)𝑒(𝑐/𝜙2)(𝑡𝑠)𝑑𝑠.(3.11) Thus, for small 𝑡 and small 𝜙𝑐/2, ̃𝜌2(1,𝑦,𝑡) is away from 0, but, as time increases, its magnitude decreases proportionally to 𝑒(𝑐/𝜙2)𝑡.

Next, we use parameters from Example 3.2 and plot them in Figure 3. In this example the conductivity 𝑘1 is much larger than that of previous case, the local gradients in the micro-model are smaller, and the solutions to the macro-model achieve a smoother profile faster.

3.2.3. Quantitative Comparison—Fine Grid in Macro-Model

Now we are ready to discuss a quantitative comparison between solutions to the micro- and macro-models. For this, we set up a family of cases with 𝜀=1/(2𝑙+1),  𝑙1. We use uniform mesh with =1=2=𝜀/20 for now. See Section 3.2.4 for 𝑖 and Section 4 for adaptive gridding.

From [11, Theorem  18, page 208] and from our plots it follows that we can actually compare the micro- and macrosolutions, as long as the characteristic functions 𝜒𝑖 are involved. While the results in [11] use two-scale convergence with 𝜀, that is, rely on special periodic test functions, we compare 𝜌𝑛𝜀, and ̃𝜌𝑛𝑖, directly. We see that it is easy to do so only in the connected region Ω1.

We define a quantity 𝑒𝑛𝜀=(𝜌𝑛𝜀,̃𝜌𝑛1,1)𝜒1 to be used in comparison. We also define, for a fixed 𝑡(0,𝑇] and some function 𝑧, 𝑧(𝑥,𝑦;𝑡),𝐿1=10||||𝑧(𝑥,0.5;𝑡)𝑑𝑥,𝑧(𝑥,𝑦;𝑡),𝐿2=10||||𝑧(𝑥,0.5;𝑡)2𝑑𝑥1/2,(3.12) which will be applied to 𝑒𝑛𝜀. Clearly ,𝑊 is not a norm but is a useful quantity of interest.

In Tables 1, 2, 3, and 4 we present 𝑒𝑛𝜀,𝑊 for different 𝜀 and 𝑊=𝐿2,𝐿1. For each case we consider convergence of 𝑒𝑛𝜀,𝑊 with 𝜀0.

# cells 𝜀 𝑒 𝑁 𝜀 , 𝐿 1 𝛼 1 𝑒 𝑁 𝜀 , 𝐿 2 𝛼 2

3 × 3 1 / 3 1 . 1 8 3 2 × 1 0 2 3 . 2 0 7 1 × 1 0 2
5 × 5 1 / 5 7 . 0 9 1 5 × 1 0 3 1 . 0 1 8 8 1 . 9 0 4 4 × 1 0 2 1 . 0 3 9 0
7 × 7 1 / 7 5 . 0 5 2 2 × 1 0 3 1 . 0 0 7 7 1 . 3 5 2 8 × 1 0 2 1 . 0 1 6 4
9 × 9 1 / 9 3 . 9 2 8 5 × 1 0 3 1 . 0 0 1 1 1 . 0 5 0 2 × 1 0 2 1 . 0 0 7 5
1 1 × 1 1 1 / 1 1 3 . 2 1 6 2 × 1 0 3 0 . 9 9 6 9 8 . 5 8 9 0 × 1 0 3 1 . 0 0 2 1

# cells 𝜀 𝑒 𝑁 𝜀 , 𝐿 1 𝛼 1 𝑒 𝑁 𝜀 , 𝐿 2 𝛼 2

3 × 3 1 / 3 1 . 1 4 4 6 × 1 0 2 3 . 0 7 1 5 × 1 0 2
5 × 5 1 / 5 7 . 0 0 6 5 × 1 0 3 1 . 0 4 0 8 1 . 8 3 8 7 × 1 0 2 0 . 9 9 5 6
7 × 7 1 / 7 5 . 1 5 4 1 × 1 0 3 1 . 0 9 5 8 1 . 3 3 6 3 × 1 0 2 1 . 0 5 4 3
9 × 9 1 / 9 4 . 1 7 0 4 × 1 0 3 1 . 1 8 6 7 1 . 0 6 8 4 × 1 0 2 1 . 1 2 3 2
1 1 × 1 1 1 / 1 1 3 . 5 7 9 4 × 1 0 3 1 . 3 1 3 1 9 . 0 4 6 0 × 1 0 3 1 . 2 0 5 8

# cells 𝜀 𝑒 𝑁 𝜀 , 𝐿 1 𝛼 1 𝑒 𝑁 𝜀 , 𝐿 2 𝛼 2

3 × 3 1 / 3 1 . 3 7 5 8 × 1 0 2 3 . 5 3 9 0 × 1 0 2
5 × 5 1 / 5 8 . 2 8 1 0 × 1 0 3 1 . 0 3 3 0 2 . 0 9 9 8 × 1 0 2 0 . 9 7 8 6
7 × 7 1 / 7 5 . 7 3 4 1 × 1 0 3 0 . 9 1 5 5 1 . 4 7 4 1 × 1 0 2 0 . 9 5 1 0
9 × 9 1 / 9 4 . 3 9 8 7 × 1 0 3 0 . 9 4 7 9 1 . 1 3 3 6 × 1 0 2 0 . 9 5 6 8

# cells 𝜀 𝑒 𝑁 𝜀 , 𝐿 1 𝛼 1 𝑒 𝑁 𝜀 , 𝐿 2 𝛼 2

3 × 3 1 / 3 1 . 1 8 2 6 × 1 0 2 3 . 2 0 4 8 × 1 0 2
5 × 5 1 / 5 7 . 0 8 7 7 × 1 0 3 0 . 9 9 7 8 1 . 9 0 2 9 × 1 0 2 0 . 9 8 0 0
7 × 7 1 / 7 5 . 0 4 9 7 × 1 0 3 0 . 9 9 2 4 1 . 3 5 1 8 × 1 0 2 0 . 9 8 4 0
9 × 9 1 / 9 3 . 9 2 8 1 × 1 0 3 1 . 0 0 0 6 1 . 0 5 0 2 × 1 0 2 0 . 9 9 5 5

From homogenization theory it is not clear what order of convergence should be expected. We observe that 𝑒𝑛𝜀,𝑊 decreases linearly with 𝜀 for both =𝐿1,𝐿2 -like quantities in all cases.

Next we discuss the dependence of the results on conductivity and other data. Consider Example 3.2 as a reference example with its corresponding Table 1. Compare it now with that from Table 2 where 𝑘3 is larger than that in the reference case to see how this influences the errors. As expected, and as shown in Example 3.3, 𝑐 changes by the same factor, and this means that the coupling between ̃𝜌1 and ̃𝜌2 is stronger which corresponds to faster flow across 𝑌3 in the micro-model. However, the influence on the convergence of the micro-macro difference, from Table 2 is rather weak.

Next consider Example 3.4 which uses a different geometry than that in the reference case, with thicker region 𝑌3. This produces a slightly smaller 𝑐 since the gradient of 𝑈 is smaller. Also, 𝜙1 and ̃𝑘1 are predictably different. The error quantity is influenced insignificantly.

The opposite effect is seen in Table 4 when 𝑌3 is selected as thinner than in the reference Example 3.2.

3.2.4. Coarse Macrogrid and Parameter Grid

In the examples presented in Section 3.2.3 we used =1=2. This choice of compatible grids is convenient for visualization purposes. However, for the idea of the macro-model to be useful, the computational complexity of the macro-model needs to be lower than that of the micro-model. Since the macro-model is a system of two equations, its lower complexity can be only achieved if the grid in the macro-model is significantly coarser than that of micro-model. Below we show that one can use 1 in the macro-model.

Example 3.6. We use data from Example 3.2 with 𝜀=0.2. Consider a fixed microgrid with 1=100 and a few cases of macrogrid with 11=21=10,50,100. Now we compare the micro- and macrosolutions 𝜌𝑛𝜀, and 𝜌1,1, see Table 5. We also compare the run time of the macro-model to that of the reference micro-model which is 120 seconds.

1 1 𝑒 𝑁 𝜀 , 𝐿 1 𝑒 𝑁 𝜀 , 𝐿 2 run time(s)

100 7 . 0 9 1 5 × 1 0 3 1 . 9 0 4 4 × 1 0 2 511
50 7 . 0 9 3 7 × 1 0 3 1 . 9 0 4 8 × 1 0 2 56
10 7 . 2 3 8 0 × 1 0 3 1 . 9 1 7 3 × 1 0 2 3

We see that, as predicted above, the macro-model run time at 𝑖= is not competitive with that of the micro-model; this is exacerbated by the overhead of various adaptive procedures to be discussed below. However, when 𝑖, the ability of the macro-model solutions to approximate those of micro-model appears reasonable and is associated with a much lower cost. This is true even when 𝑖 is by a factor of 10 coarser than resulting in a computational time decreased by two orders of magnitude.

Clearly, the macro-model can run much faster than the micro-model especially when adaptive nonuniform grids with 12 are implemented, see Section 4.

Influence of Parameter Grid
Last we address the effect of the computed parameters 𝑐, ̃𝑘1 on the closeness of the macro-model solution to that of micro-model. These coefficients are precomputed as discussed in Section 3.1.1.

Example 3.7. Let 𝜀=1/3, and let the micro-model grid be 1=60. Use 11=21=60. Compute numerical parameters 𝑐𝑐, ̃𝑘1,𝑘1 by approximating solutions to (3.4a), (3.4b), (3.4c), and (3.4d) and (3.3a), (3.3b), and (3.3c), respectively.
Example 3.7 and Table 6 show that the grid used for computing these parameters only mildly affects the quality of the solution, thus one can use coarse grid with confidence.

𝑐 1 𝑐 𝑐 𝑒 𝑁 𝜀 , 𝐿 1 𝑒 𝑁 𝜀 , 𝐿 2

10 1 . 9 0 0 0 × 1 0 3 3 . 6 2 5 1 × 1 0 2 5 . 9 0 2 6 × 1 0 2
50 1 . 8 3 7 0 × 1 0 3 3 . 6 2 5 1 × 1 0 2 5 . 9 0 2 6 × 1 0 2
100 1 . 8 2 9 2 × 1 0 3 3 . 6 2 5 1 × 1 0 2 5 . 9 0 2 6 × 1 0 2
200 1 . 8 2 5 8 × 1 0 3 3 . 6 2 5 1 × 1 0 2 5 . 9 0 2 6 × 1 0 2

𝑘 1 ̃ 𝑘 1 ( 1 , 1 ) 𝑒 𝑁 𝜀 , 𝐿 1 𝑒 𝑁 𝜀 , 𝐿 2

10 4 . 6 3 0 6 × 1 0 1 3 . 5 3 9 2 × 1 0 2 5 . 7 8 3 5 × 1 0 2
50 4 . 5 1 4 1 × 1 0 1 3 . 6 2 2 2 × 1 0 2 5 . 8 9 8 9 × 1 0 2
100 4 . 5 0 4 6 × 1 0 1 3 . 6 3 0 1 × 1 0 2 5 . 9 0 9 3 × 1 0 2
200 4 . 5 0 0 9 × 1 0 1 3 . 6 3 3 2 × 1 0 2 5 . 9 1 3 4 × 1 0 2

4. Error Estimates and Two-Grid Adaptivity

We present now the details of discrete formulation of (2.4) using independent grids for each component and adaptive gridding. The adaptive two-grid algorithm further reduces the computational cost of the macro-model. The formalism of the two-grid solution connects loosely to [37, 38] where their benefit was considered for a nonlinear scalar equation.

In what follows we assume that ̃𝑘1̃𝑘2, that is, that the first component varies in space faster than the second. As a special case, this includes the case ̃𝑘2=0. We consider two triangulations 𝒯𝑖, 𝑖=1,2, which will be used for ̃𝜌𝑖, respectively. In principle, these can be chosen independently. In fact, we allow the triangulations to vary in time and thus consider 𝒯𝑛𝑖 and the associated spaces 𝑉𝑛𝑖 as in (2.7). To avoid the loss of accuracy due to excessive interpolation and intergrid projections and because ̃𝑘1̃𝑘2, we assume that 𝒯𝑛1 is a refinement of the partition 𝒯𝑛2.

We need intergrid operators to handle two components that live on separate grids. Let Π𝑉𝑛2𝑉𝑛1 be the interpolation operator and Π𝑉𝑛1𝑉𝑛2 the 𝐿2 projection defined by Π𝜙1,𝜓2𝜙=1,Π𝜓2,𝜓2𝑉2.(4.1)

Now we define the discrete solutions. At each time step 1𝑛𝑁, we seek ̃𝜌𝑛𝑖,𝑖𝑉𝑛𝑖, 𝑖=1,2, which satisfy the discrete problem, that is, (2.4) restricted to the finite dimensional subspaces 𝑉𝑛𝑖 so that, for all 𝜓1𝑉𝑛1, for all 𝜉2𝑉𝑛2𝜙1̃𝜌𝑛1,1̃𝜌𝑛11,1Δ𝑡𝑛,𝜓1+̃𝑘1̃𝜌𝑛1,1,𝜓1+𝑐̃𝜌𝑛1,1Π̃𝜌𝑛2,2,𝜓1=𝑓,𝜓1,𝜙2̃𝜌𝑛2,2̃𝜌𝑛12,2Δ𝑡𝑛,𝜉2+̃𝑘2̃𝜌𝑛2,2,𝜉2+𝑐̃𝜌𝑛2,2Π̃𝜌𝑛1,1,𝜉2=0,̃𝜌01,1,̃𝜌02,2=1̃𝜌1,0,2̃𝜌2,0inΩ.(4.2) If ̃𝑘2=0, the system (4.2) is modified appropriately so that ̃𝜌2(,𝑡)𝐿2(Ω) instead of 𝐻1𝐷(Ω), and we do not impose boundary conditions on this component. In fact, instead of (2.7) we define 𝑉𝑛2=𝑣𝐶Ω𝑇𝒯𝑛,𝑣𝑇𝒫𝑘(𝑇).(4.3) (It holds that 𝑣𝐶(Ω) but it is not necessary). It is straightforward that (4.2) is uniquely solvable.

4.1. A Priori Error Estimate

We have the following convergence result proved for the error in energy norm. In what follows 1𝑛𝑁 and 𝑖=1,2. We denote 𝑒𝑖,𝑖(𝑡𝑛)=̃𝜌𝑛𝑖,𝑖̃𝜌𝑛𝑖 and define the energy norm for the product space ||||𝑧,𝑤2𝑡𝑛𝜙=1𝑧𝑡𝑛2+𝜙2𝑤𝑡𝑛2+𝑛𝑚=1Δ𝑡𝑚̃𝑘1||𝑧𝑡𝑚||21+̃𝑘2||𝑤𝑡𝑚||21.(4.4)

Assuming that ̃𝜌1,̃𝜌2 is sufficiently smooth we have the following a-priori estimate proven in the appendix.

Theorem 4.1. Let ̃𝜌1,̃𝜌2,{̃𝜌𝑛1,1,̃𝜌𝑛2,2}𝑁𝑛=1 be the smooth solutions of (2.4) and (4.2), respectively. Then one has ||𝑒1,1,𝑒2,2||𝑡𝑛̃𝑘11/2𝐶11+̃𝑘21/2𝐶22+𝐶3max𝑚=1𝑛Δ𝑡𝑚+higherorderterms(h.o.t.),(4.5) where 𝐶1, 𝐶2, and 𝐶3 are independent of .
Furthermore, if ̃𝑘2=0, then ||𝑒1,1,𝑒2,2||𝑡𝑛̃𝑘11/2𝐶11+𝐶222+𝐶3max𝑚=1𝑛Δ𝑡𝑚+higherorderterms(.𝑜.𝑡.).(4.6)

Now we notice that (4.5) suggests the following choice of grid parameters 1, 2. If ̃𝑘1̃𝑘2, then to balance the components of the error one can use 12. Similarly, if ̃𝑘2=0, then (4.6) implies that 2 can be chosen to be of the order of 11/2. The use of coarse grid 21 for the second component reduces the size of the linear system to be solved and decreases the computational time of the macro-model.

Also, the a-priori estimates (4.5), (4.6) are global. One can further reduce the computational cost while maintaining accuracy by using local grid adaptivity guided by a posteriori estimators. This is pursued below.

4.2. A-Posteriori Error Estimates

Estimators for time-dependent problems can be defined in many ways including the now classical space-time element and adjoint approaches [27, 28]. In this paper we follow the residual estimator framework extending [15, 29] to the double-diffusion system using the ideas in [13, 39, 40] originally formulated for an elliptic system.

The estimator 𝜂𝑛=(𝑛𝑚=1[Δ𝑡𝑚𝑆2𝑚+𝑇2𝑚])1/2 for (4.2) is composed of the temporal part 𝑇𝑛 which adapts the time stepping and the spatial part 𝑆𝑛 which guides the spatial grid adaptivity. We define 𝑇2𝑛=Δ𝑡𝑛3̃𝑘11/2̃𝜌𝑛1,1̃𝜌𝑛11,12+̃𝑘21/2̃𝜌𝑛2,2̃𝜌𝑛12,22+Δ𝑡𝑛3𝑐1/2̃𝜌𝑛1,1̃𝜌𝑛11,1𝑐1/2̃𝜌𝑛2,2̃𝜌𝑛12,22,(4.7) and 𝑆2𝑛=𝑆1,𝑛+𝑆2,𝑛, where 𝑆𝑖,𝑛=𝑇𝑖𝒯𝑖𝑆𝑛𝑖,𝑇𝑖, and where, as usual (see [23]), one defines 𝑆𝑛𝑖,𝑇𝑖=𝜃2𝑖,𝑇𝑖𝑅𝑛𝑖,𝑇𝑖2+12𝐸𝑇𝑖𝛾2𝑖,𝐸𝑅𝑛𝑖,𝐸2.(4.8) The element and edge residuals in 𝑆𝑛𝑖,𝑇𝑖,𝑖=1,2, are given by 𝑅𝑛𝑖,𝑇𝑖=𝑓𝑖𝜙𝑖̃𝜌𝑛𝑖,𝑖̃𝜌𝑛1𝑖,𝑖Δ𝑡𝑛̃𝑘+𝑖̃𝜌𝑛𝑖,𝑖+(1)𝑖𝑐̃𝜌𝑛1,1̃𝜌𝑛2,2,𝑅(4.9)𝑛𝑖,𝐸=̃𝑘𝑖𝜕𝜈̃𝜌𝑛𝑖,𝑖𝐸̃𝑘if𝐸𝜕Ω,𝑖𝜕𝜈̃𝜌𝑛𝑖,𝑖if𝐸Γ𝑁,0,otherwise.(4.10) Here [𝑧]𝐸 denotes the jump of the flux 𝑧 across the edge 𝐸 of an element. Usually 𝑓1𝑓, 𝑓20. Also, if ̃𝑘2=0, then 𝑅𝑛2,𝐸0.

The scaling constants 𝜃𝑖,𝑇,𝛾𝑖,𝐸 take into account the contribution of the exchange term 𝑐(̃𝜌1̃𝜌2) and are defined so that the estimators remain robust when the coefficients of the problem change substantially: 𝜃1,𝑇=min𝑇1̃𝑘11/2𝑐,max1/2,𝑇1̃𝑘21/2,𝑇1𝒯𝑛1𝑛1𝜃,(4.11)2,𝑇=min𝑇2̃𝑘21/2𝑐,max1/2,𝑇2̃𝑘11/2,𝑇2𝒯𝑛2𝑛2𝛾,(4.12)𝑖,𝐸=2𝐸1/2𝜃𝑖,𝐸,𝑖=1,2.(4.13) This a posteriori estimator works well with two-grid discretizations as shown in the examples to follow. It is well formulated also for ̃𝑘20 when ̃𝜌2𝐿2(Ω). We have the following result on reliability of the estimator proven in the appendix.

Theorem 4.2. Let ̃𝜌1,̃𝜌2, {̃𝜌𝑛1,1,̃𝜌𝑛2,2}𝑁𝑛=1 be the solution of (1.2a) and (1.2b), (2.3) and (4.2), respectively. Then ||𝑒1,1,𝑒2,2||𝑡𝑛𝜂𝑛+h.o.t.;𝑛1(4.14)

4.3. Adaptive Two-Grid Discretization and Implementation

Consider a fixed time step 𝑛=𝑛𝑜 for which some triangulation is chosen, the solution is found, and the spatial error indicators 𝑆𝑖,𝑛 are computed. To apply the two-grid spatial adaptivity we use the following refinement algorithm.

Adaptive Two-Grid Algorithm 𝓐
(1)Select the triangulation(s) to be refined: If 𝑆1,𝑛𝑜>3𝑆2,𝑛𝑜, refine only 𝒯𝑛𝑜1. If 𝑆2,𝑛𝑜>3𝑆1,𝑛𝑜, refine only 𝒯𝑛𝑜2. Otherwise, refine both. (2)In 𝒯𝑖, 𝑖=1,2 selected in Step 1, refine any 𝑇𝒯,𝑖 for which 𝑆𝑛𝑜𝑖,𝑇0.5max𝑇𝒯𝑖𝑆𝑛𝑜𝑖,𝑇. (3)Enforce the requirement that 𝒯𝑛1 is a refinement of 𝒯𝑛2 by adding extra elements as needed.
The steps in 𝒜 should be repeated a certain number 𝑁𝒜 of times until an ideal grid is found. See [41] for analysis of whether the iterative process of refinement and coarsening is, in general, convergent.

Example of Two-Grid Adaptivity
Now we show an example on how this adaptive algorithm works for the double-diffusion system. We use data and setup from Example 3.2, with 𝑇=0.02 and uniform time stepping with Δ𝑡=104. Other data is as in Section 3.2.1.
Consider a fixed 𝑛0=200 with a uniform triangulation 𝒯𝑛𝑜1=𝒯𝑛𝑜2 with 1=2=0.1. Then apply 𝑁𝐴=6 times the steps in 𝒜, see the solution and the refined meshes in Figure 4. The process works as expected: since at the time 𝑡𝑛0 the solution has a high gradient near 𝑥=1, the refinement occurs there. The refinement affects the grid for the first component only, because the gradients of the second component are not included due to ̃𝑘2=0. This can be compared to the a-priori estimate in Theorem 4.1 from which we know the convergence is 𝑂(1+22+Δ𝑡𝑛).
In Table 7 we provide the details on 𝒯𝑛𝑜𝑖 and compare the effectiveness of the local two-grid refinement by algorithm 𝒜 with uniform grid refinement. With adaptive refinement we get 𝑆𝑛𝑜=0.35834 with 435+121=556 unknowns, while the uniform refinement needs 441+441=882 unknowns to achieve comparable 𝑆𝑛𝑜=0.39221.

Uniform refinement
𝑆 𝑛 𝑜 # elem 𝒯 𝑛 𝑜 1 # elem 𝒯 𝑛 𝑜 1 d i m ( 𝑉 𝑛 𝑜 1 ) d i m ( 𝑉 𝑛 𝑜 2 )

Initial mesh 0 . 8 1 8 0 3 200 200 121121
1st refinement 0 . 3 9 2 2 1 800 800 441441

Adaptive refinement
𝑆 𝑛 𝑜 # elem 𝒯 𝑛 𝑜 1 # elem 𝒯 𝑛 𝑜 2 d i m ( 𝑉 𝑛 𝑜 1 ) d i m ( 𝑉 𝑛 𝑜 1 )

Initial mesh 0 . 8 1 8 0 200 200 121121
1st level 0 . 8 0 3 1 240 200 141121
2nd level 0 . 7 0 7 6 292200 167121
3rd level 0 . 6 4 9 9 316200 179121
4th level 0 . 5 5 2 5 462200 254121
5th level 0 . 4 2 0 5 720200385121
6th level 0 . 3 5 8 3 820200 435121

Last, we describe the process with which 𝒯𝑛𝑖,  𝑖=1,2, may vary between time steps. For a fixed 𝑛 denote by 𝒯𝑛𝑖 the final triangulation obtained by the adaptive algorithm 𝒜. For the new time step 𝑛+1, we use 𝒯𝑛𝑖 as the initial triangulation. If the algorithm 𝒜 suggests that 𝒯𝑛+1𝑖 is modified, we need to project ̃𝜌𝑛𝑖,𝑖 to the new grid so it can be used as initial data for the new step, but ̃𝜌𝑛𝑖,𝑖 differs from their interpolation only in the elements where the triangulation is coarsened. Another way to deal with different triangulations 𝒯𝑛+1𝑖, 𝒯𝑛𝑖 is to compute the a posteriori error estimator in a common refinement of the two triangulations 𝒯𝑛+1𝑖, 𝒯𝑛𝑖 [29, 42], but this will not be pursued further.

The decision to use two grids implies that we have to interpolate between the finite element spaces 𝑉𝑛1 and 𝑉𝑛2, see (4.1). Suppose that {𝜁𝑖,𝑗}𝑗,  𝑖=1,2, is a basis for 𝑉𝑛𝑖. Then the operator Π in (4.1) is represented by the matrix 𝕀1,2 whose components are given by 𝕀1,2(𝑗,𝑘)=(𝜁1,𝑗,𝜁2,𝑘).
Consider now and compare the computational cost associated with one-grid and two-grid approaches. To solve (4.2) with one-grid 𝑉𝑛1 we need to assemble the stiffness matrix 𝑆1,1̃𝑘(𝑗,𝑘)=(1𝜁1,𝑗,𝜁1,𝑘), the mass matrix 𝑀1,1(𝑗,𝑘)=(𝜁1,𝑗,𝜁1,𝑘), and the block matrix 𝐵one-grid=Δ𝑡𝑆1,1+𝜙1𝑀+𝑐Δ𝑡1,1𝑐Δ𝑡𝑀1,1𝑐Δ𝑡𝑀1,1𝜙2𝑀+𝑐Δ𝑡1,1.(4.15) At each time step we solve the linear system with 𝐵onegrid; this is referred to as 𝕊one-grid.
To solve (4.2) with two grids and two spaces 𝑉𝑛1,𝑉𝑛2, we need to assemble the stiffness matrix 𝑆1,1̃𝑘(𝑗,𝑘)=(1𝜁1,𝑗,𝜁1,𝑘), the mass matrices 𝑀1,1(𝑗,𝑘)=(𝜁1,𝑗,𝜁1,𝑘), 𝑀2,2(𝑗,𝑘)=(𝜁2,𝑗,𝜁2,𝑘), the interpolation matrix 𝕀1,2(𝑗,𝑘)=(𝜁1,𝑗,𝜁2,𝑘), and the block matrix 𝐵two-grid=Δ𝑡𝑆1,1+𝜙1𝑀+𝑐Δ𝑡1,1𝑐Δ𝑡𝕀𝑇1,2𝑐Δ𝑡𝕀1,2𝜙2𝑀+𝑐Δ𝑡2,2.(4.16) To finish we solve the linear system which we refer to as 𝕊two-grid.

Suppose now we solve (4.2) and wish to maintain 𝑆𝑛𝛿 for some tolerance 𝛿 over 1,000 time steps. With data as in Section 3.1.2 we run the simulations using one grid and find that we need 1=2=1/80. Using the two-grid approach we only need 1=1/80,2=1/20.

The computational time needed to assemble the matrices and solve the systems in our MATLAB implementation is displayed in Table 8. Clearly the assembly process takes more time for two-grid case than for one-grid discretization. However, the economy in the cost of solving the linear system makes up for that cost.

One-grid with 1 = 2 = 1 / 8 0
Matrix Cost (seconds)

𝑆 1 , 1 + 𝑀 1 , 1 3.6547
𝐵 o n e - g r i d 0.0123
Solve 𝕊 o n e - g r i d 1,000 times0.2761 × 1,000


Two-grid with 1 = 1 / 8 0 , 2 = 1 / 2 0
Matrix Cost (seconds)

𝑆 1 , 1 + 𝑀 1 , 1 3.6547
𝑀 2 , 2 0.0438
𝕀 1 , 2 0.2121
𝐵 t w o - g r i d 0.0063
Solve 𝕊 t w o - g r i d 1,000 times 0.1030 × 1,000


5. Conclusions

We compared the solutions to the micro-model to those of the macro-model and have shown that the latter is a good approximation to the former. This remains true also when a very coarse computational grid is used. Moreover, we established a linear convergence rate in function of the periodicity parameter 𝜀 in a certain quantity of interest which further shows that the double-diffusion model is an excellent approximation to the micro-model, at least for the considered scenarios.

To make the double-diffusion model computationally efficient, we proposed an algorithm for local grid adaptivity which allows each component to live on its own grid. The grid refinement is guided by an a posteriori error estimator for which we proved theoretical results.


A. Proof of Theorem 4.1

Proof. The proof follows the techniques presented in [25, 26] for a scalar PDE. We present an outline of the proof for the double-diffusion system. By (4.1), if 𝒯𝑛1 is a refinement of 𝒯𝑛2, we can rewrite (4.2) as 𝜙1̃𝜌𝑛1,1̃𝜌𝑛11,1Δ𝑡𝑛,𝜓1+̃𝑘1̃𝜌𝑛1,1,𝜓1+𝑐̃𝜌𝑛1,1̃𝜌𝑛2,2,𝜓1=𝑓,𝜓1,𝜙2̃𝜌𝑛2,2̃𝜌𝑛12,2Δ𝑡𝑛,𝜉2+̃𝑘2̃𝜌𝑛2,2,𝜉2+𝑐̃𝜌𝑛2,2̃𝜌𝑛1,1,𝜉2=0.(A.1) Next, for ̃𝑘20, as in [26] we define the elliptic projection 𝑃̃𝜌1,𝑃̃𝜌2 of ̃𝜌1,̃𝜌2 into 𝑉1×𝑉2 via 𝑃̃𝜌𝑖̃𝜌𝑖,𝜒=0,𝜒𝑉𝑖,𝑖=1,2.(A.2) Applying (A.2) for 𝑖=1,2 into the weak formulation (2.4), with arbitrary 𝜓,𝜉𝐻1𝐷(Ω) we arrive at 𝜙1𝜕𝑃̃𝜌1+̃𝑘𝜕𝑡,𝜓1𝑃̃𝜌1+𝑐,𝜓̃𝜌1̃𝜌2𝜙,𝜓=(𝑓,𝜓)+1𝜕𝑃̃𝜌1̃𝜌1,𝜙𝜕𝑡,𝜓2𝜕𝑃̃𝜌2+̃𝑘𝜕𝑡,𝜉2𝑃̃𝜌2+𝑐,𝜉̃𝜌2̃𝜌1=𝜙,𝜉2𝜕𝑃̃𝜌2̃𝜌2.𝜕𝑡,𝜉(A.3)
Next we define 𝜕𝑛𝑤=(𝑤(𝑡𝑛