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

Abstract

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, [1–3]. 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−∇⋅(𝑘∇𝜌)=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 fractures𝜙₁𝜕𝜌̃₁𝜕𝑡−∇⋅𝑘̃¹𝜙∇𝜌̃₁+𝑐𝜌̃₁−𝜌̃₂=𝑓,inΩ,𝑡>0,(1.2a)₂𝜕𝜌̃₂𝜕𝑡−∇⋅𝑘̃²∇𝜌̃₂+𝑐𝜌̃₂−𝜌̃₁=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(𝜌̃₂−𝜌̃₁), and the coefficients 𝑘̃_𝑖, 𝜙_𝑖 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 𝑘̃₂=0 known as the Warren-Root model [6]𝜙₁𝜕𝜌̃₁𝜕𝑡−∇⋅𝑘̃¹𝜙∇𝜌̃₁+𝑐𝜌̃₁−𝜌̃₂=𝑓,inΩ,𝑡>0,(1.3a)₂𝜕𝜌̃₂𝜕𝑡+𝑐𝜌̃²−𝜌̃₁=0,inΩ,𝑡>0,(1.3b)is a popular way of modeling of subscale diffusion accompanying advection and adsorption, see [7–10].

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 𝑘̃₂=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 0≤𝑘̃₂≪𝑘̃₁, it is natural to suggest that in numerical simulation 𝜌̃₁ and 𝜌̃₂ 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 𝑘̃₂≈0 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 [14–16] 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 Ω⊂ℝ^𝑑, 𝑑=2,3, be a bounded polygonal region with piecewise smooth boundary 𝜕Ω=Γ_𝐷∪Γ_𝑁, where Γ_𝐷, Γ_𝑁 are disjoint regions of the boundary 𝜕Ω where Dirichlet and Neumann boundary conditions are prescribed, respectively.

For any subset 𝜔⊆Ω, we use the standard notation for Lebesgue 𝐿²(𝜔) and Sobolev spaces 𝐻^𝑘(𝜔), 𝑘∈ℕ. These are equipped with the usual seminorms |⋅|_𝑘,𝜔, norms ‖⋅‖_𝑘,𝜔∶=‖⋅‖_𝐻^𝑘_(𝜔) and the usual scalar product (s) (𝑓,𝑔)_𝜔∶=(𝑓,𝑔)_𝐿²_(𝜔). We denote ‖𝑓‖_𝜔∶=(𝑓,𝑓)^𝜔1/2. If 𝜔=Ω, the subscript 𝜔 will be omitted. We also let 𝐻^1𝐷(Ω)∶={𝑣∈𝐻¹(Ω)∶𝑣|_Γ𝐷=0} under the standard norm in 𝐻¹(Ω), ‖𝑣‖₁∶=(‖𝑣‖²+‖∇𝑣‖²)^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 ‖𝑣‖_𝐿²_{(0,𝑇;𝑆)}∶=(∫^𝑇0‖𝑣‖^2𝑆)^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),𝑥∈Ω 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=𝜌𝜌=₀,𝜌_𝐷inΩ×{0},𝑘∇𝜌⋅_→,onΓ_𝐷,𝑛=0,onΓ_𝑁.(2.1) For simplicity we assume that 𝜌_𝐷≡0 and that the flow is driven by 𝑓. However, in simulation examples we use 𝑓≡0, 𝜌_𝐷≢0.

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

Find 𝜌∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐻^1𝐷(Ω))  so that𝜙𝜕𝜕𝜌𝑡,𝜓+(𝑘∇𝜌,𝜓)=(𝑓,𝜓),∀𝜓∈𝐻^1𝐷(Ω).(2.2) For any 𝜌₀∈𝐿²(Ω), 𝑓∈𝐶([0,𝑇];𝐿²(Ω)), there is a unique solution 𝜌∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐻^1𝐷(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 𝜙₁, 𝜙₂, 𝑘̃₁, 𝑘̃₂, and 𝑐∈𝐿^∞(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 𝑘̃₂≡0.

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 𝑘̃₂>0, we assume the following initial and boundary conditions defined for 𝑖=1,2: 𝜌̃_𝑖=𝜌̃_𝑖𝜌̃_𝑖,0,inΩ×{0},𝑘̃_𝑖=𝜌_𝐷,onΓ_𝐷,∇𝜌̃_𝑖⋅^→𝑛=0,onΓ_𝑁.(2.3) As before, we assume that 𝜌_𝐷≡0 for now.

Let 𝜌̃_𝑖(⋅,0)=𝜌̃_𝑖,0∈𝐿²(Ω), 𝑓∈𝐶([0,𝑇],𝐿²(Ω)), 𝑢_𝐷≡0. The variational form of (1.2a), (1.2b), and (2.3) is as follows:

Find 𝜌̃_𝑖∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐻^1𝐷(Ω)),  𝑖=1,2, such that𝜙¹𝜕𝜌̃₁𝜕𝑡,𝜓+𝑘̃¹𝜙∇𝜌̃₁,∇𝜓+𝑐𝜌̃₁−𝜌̃₂,𝜓=(𝑓,𝜓),∀𝜓∈𝐻^1𝐷(Ω),₂𝜕𝜌̃₂𝜕𝑡,𝜉+𝑘̃²∇𝜌̃₂,∇𝜉+𝑐𝜌̃₂−𝜌̃₁,𝜉=0,∀𝜉∈𝐻^1𝐷(Ω).(2.4) The well posedness of this system (2.4) follows immediately by showing that the operator ℬ∶𝒟⊂𝐻^1𝐷(Ω)×𝐻^1𝐷(Ω)→𝐿²(Ω)×𝐿²(Ω)ℬ=⎡⎢⎢⎣−𝜕𝑘̃¹−𝜕𝑐𝐼+𝑐𝐼−𝜕𝑘−̃₂𝑐𝐼𝜕+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 −𝜕𝑘̃_𝑖𝜕, 𝐼.

Now, if 𝑘̃₂≡0, we do not impose boundary conditions on 𝜌̃₂. The variational formulation of (3) and (2.3), reads

Find 𝜌̃₁∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐻^1𝐷(Ω)),  𝜌̃₂∈𝐶([0,𝑇];𝐿²(Ω))∩𝐿²(0,𝑇;𝐿²(Ω))   such that𝜙¹𝜕𝜌̃₁𝜕𝑡,𝜓+𝑘̃¹𝜙∇𝜌̃₁,∇𝜓+𝑐𝜌̃₁−𝜌̃₂,𝜓=(𝑓,𝜓),∀𝜓∈𝐻^1𝐷(Ω),₂𝜕𝜌̃₂𝜕𝑡,𝜉+𝑐𝜌̃²−𝜌̃₁,𝜉=0,∀𝜉∈𝐿²(Ω).(2.6) The well posedness of (2.6) follows similarly from the Hille-Yosida theorem. Also, 𝜌̃₂ has the minimum of regularity of its initial data and of 𝜌̃₁.

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 [𝑡_𝑛−1,𝑡_𝑛], 1≤𝑛≤𝑁, such that 0=𝑡₀<𝑡₁<⋯<𝑡_𝑁=𝑇. We denote Δ𝑡_𝑛=𝑡_𝑛−𝑡_𝑛−1 and 𝑧(𝑡_𝑛)=𝑧^𝑛 for any function 𝑧.

For spatial discretization, we adopt standard finite element nomenclature that can be found in textbooks such as [23–25]. At each time step 𝑛, we denote by 𝒯^𝑛ℎ, ℎ>0, a family of admissible and shape-regular partitions of Ω into a finite number of elements. For any element 𝑇∈𝒯^𝑛ℎ, we let ℎ_𝑇 be the diameter of 𝑇 and denote ℎ=max_{𝑇∈𝒯ℎ}ℎ_𝑇. Also, ℰ^𝑛ℎ is the set of all edges of 𝒯^𝑛ℎ. For any edge 𝐸∈ℰ^𝑛ℎ, we let ℎ_𝐸 denote the diameter of the edge 𝐸. Finally, we denote by 𝒫_𝑘(𝑇) the space of polynomials of degree 𝑘 in ℝ^𝑑.

Numerical Approximation of Micro-Model
At each time step 𝑛, we define 𝑉^𝑛ℎ=𝑣_ℎ∈𝐶Ω∶∀𝑇∈𝒯^𝑛ℎ,𝑣_ℎ∣_𝑇∈𝒫_𝑘(𝑇),𝑣∣_Γ𝐷=0,(2.7) and seek an approximation 𝜌^𝑛ℎ∈𝑉^𝑛ℎ such that for all 𝜓_ℎ∈𝑉^𝑛ℎ𝜙𝜌^𝑛ℎ−𝜌^ℎ𝑛−1Δ𝑡_𝑛,𝜓_ℎ+𝑘∇𝜌^𝑛ℎ𝜌,∇𝜓_ℎ=𝑓,𝜓_ℎ,(2.8a)_0ℎ=ℐ_ℎ𝜌₀inΩ.(2.8b)Here ℐ_ℎ denotes an interpolation or projection operator into 𝑉^𝑛ℎ.
It is standard [20] that there is a unique solution 𝜌^𝑛ℎ∈𝑉^𝑛ℎ for (2.8a) at each 𝑛, 1≤𝑛≤𝑁. 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, 27–29]; 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 𝜌̃_𝑖 can be associated with its own grid 𝒯^𝑛ℎ_𝑖; 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 Ω_𝑖, 𝑖=1,…,3, see Figure 1, with positive constants (𝜙_𝑖,𝑘_𝑖)^3𝑖=1 so that 𝜙𝑘((𝑥𝑥))==𝜙𝑘₁₁𝜒₁(𝑥)+𝜙₂𝜒₂(𝑥)+𝜙₃𝜒₃(𝑥),𝜒₁(𝑥)+𝑘₂𝜒₂(𝑥)+𝑘₃𝜒₃(𝑥).(3.1) Here 𝜒_𝑖 denotes the characteristic function of Ω_𝑖.

Figure 1 

Periodic heterogeneous media. (a) Ω1: white, Ω2: grey, and Ω3: dark grey. (b) Zoom of one cell 𝑌: 𝑌1: white, 𝑌2: grey, and 𝑌3: dark grey. (c) 𝑌3 in grey with the internal and external boundaries Γ2,3,Γ1,3, respectively. (d) Illustration of three regions as in Example 3.1, where Ω2 is delimited by the white boundaries, and Ω3 is the region between by the black and the white boundaries.

An accurate numerical approximation to (1.1) with (3.1) requires that the grid 𝒯_ℎ is very fine so that it lines up with the interfaces 𝜕Ω_𝑖∩𝜕Ω_𝑘. 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 𝜙_𝑖,𝑘̃_𝑖 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_ℎ.

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 𝑘₁≫𝑘₂ and/or 𝑘₁≫𝑘₃.

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 Ω_𝑖 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 𝜙₃≈0 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 [31–35] 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 𝑘̃₂=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 𝜙₁, 𝜙₂, 𝑘̃₁, 𝑘̃₂, 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 𝜙_𝑖, 𝑘_𝑖 and on the geometry of Ω^𝜀. We follow closely [11].

First we formalize the notion of periodicity and heterogeneity in Ω. Without loss of generality we assume that Ω^𝜀1 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)^𝑑⊂ℝ^𝑑 be divided into three distinct regions 𝑌₁, 𝑌₂, and 𝑌₃, such that 𝜕𝑌₁∩𝜕𝑌₂=∅ and 𝑌=∪^3𝑖=1𝑌_𝑖 as illustrated by Figure 1(b). Denote by 𝐶_𝑖 the characteristic function of 𝑌_𝑖, 𝑖=1,2,3, extended 𝑌-periodically to all ℝ^𝑑. We assume that the domains {𝑥∈ℝ^𝑑∶𝐶_𝑖(𝑥)=1}, 𝑖=1,2,3, have a smooth boundary. Also, we define the 𝜀-periodic characteristic functions 𝐶^𝜀𝑖(𝑥)=𝐶_𝑖(𝑥/𝜀), 𝑖=1,2,3,  𝑥∈Ω. Thus, for a fixed 𝜀, Ω is subdivided into three distinct regions Ω^𝜀𝑖={𝑥∈Ω∶𝐶^𝜀𝑖(𝑥)=1}, 𝑖=1,2,3. Another subdivision of Ω is into 𝑂(𝜀^−𝑑)-periodic cells.

Macroporosities 𝜙_𝑖
From [11, equation 16, page 206] we have 𝜙_𝑖=∫_𝑌𝑖𝜙_𝑖(𝑦)𝑑𝑦.

Macroconductivities 𝑘̃_𝑖
From [11, equations 16, 14, page 205-206] we have 𝑘̃_𝑝(𝑖,𝑗)=_𝑌𝑝𝑘_𝑝(𝑦)^→𝑒_𝑖+∇𝑊^𝑝𝑖(𝑦)⋅^→𝑒_𝑗+∇𝑊^𝑝𝑗(𝑦)𝑑𝑦,𝑝=1,2;𝑖,𝑗=1,…𝑑,(3.2) where 𝑊^𝑝𝑖, for 𝑝=1,2;𝑖=1,…,𝑑, is the solution to an auxiliary PDE∇⋅𝑘_𝑝^→𝑘𝑒_𝑖+∇𝑊^𝑝𝑖(𝑦)=0,in𝑌_𝑝,(3.3a)_𝑝^→𝑊𝑒_𝑖+∇𝑊^𝑝𝑖(𝑦)⋅^→𝑛_𝑝=0,onΓ_𝑝,3,(3.3b)_𝑝𝑖,𝑘_𝑝∇𝑊^𝑝𝑖⋅^→𝑛_𝑝areperiodic.(3.3c)One can prove [1, 3, 11] that 𝑘̃_𝑝 is a symmetric matrix.
Now consider 𝑑=2. Here the region Ω^𝜀2 cannot be connected. Thus, the boundary condition (3.3c) does not apply and (3.3a), (3.3b), and (3.3c) admit the trivial solution ^→𝑒_𝑖+∇𝑊^2𝑖(𝑦)≡0 for 𝑖=1,2. Consequently, 𝑘̃₂ 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(𝜌̃₁−𝜌̃₂). To compute 𝑐, we consider the solution 𝑈 of−∇⋅𝑈𝑘=₃∇1𝑈,o=n0Γ,_1,3in𝑌₃,(3.4a)𝑈=0,onΓ_2,3,(3.4b)𝑈,𝑘₃,(3.4c)∇𝑈⋅^→𝑛₃areperiodiconΓ_3,3.(3.4d)The parameter 𝑐 see [11, equation 19, page 209] is given by 𝑐=_Γ1,3𝑘₃∇_𝑦𝑈⋅^→𝑛₃𝑑𝑠.(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 𝜙₃≈0.

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 ℎ^𝑘−1₁=40 and ℎ^𝑐−1=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 𝑘̃_𝑖, 𝑐 and provide examples of their calculations for various values of 𝑘₃ and choices of geometry of 𝑌.

In all examples 𝑑=2 hence 𝑘̃₂=⁰⁰⁰⁰. Also, in all examples 𝑘₁≡const and the geometry of 𝑌 is axisymmetric, and thus 𝑘̃₁ is a scalar constant.

Example 3.1. Here 𝑌₂=(0.3,0.7)²,  𝑌₃=(0.2,0.8)²⧵𝑌₂,  𝜙=1𝜒₁+10⁻⁴𝜒₂,  and 𝑘=1𝜒₁+10⁻⁴𝜒₂+10⁻⁷𝜒₃. We obtain 𝜙₁=0.64,𝜙₂=1.6×10⁻⁵,𝑘̃₁=0.4519,𝑐=1.8634×10⁻⁶.(3.6)

Example 3.2. Let 𝑌₂=(0.3,0.7)²,  𝑌₃=(0.2,0.8)²⧵𝑌₂. 𝜙=1𝜒₁+10⁻⁴𝜒₂, and  𝑘=1𝜒₁+10⁻¹𝜒₂+10⁻⁴𝜒₃. We obtain 𝜙₁=0.64,𝜙₂=1.6×10⁻⁵,𝑘̃₁=0.4519,𝑐=1.8634×10⁻³.(3.7)

Example 3.3. Here 𝑌₂=(0.3,0.7)²,  𝑌₃=(0.2,0.8)²⧵𝑌₂,  𝜙=1𝜒₁+10⁻⁴𝜒₂,  and 𝑘=1𝜒₁+10⁻¹𝜒₂+10⁻²𝜒₃. We obtain 𝜙₁=0.64,𝜙₂=1.6×10⁻⁵,𝑘̃₁=0.4519,𝑐=1.8634×10⁻¹.(3.8)

Example 3.4. Let 𝑌₂=(0.3,0.7)²,  𝑌₃=(0.1,0.9)²⧵𝑌₂,  𝜙=1𝜒₁+10⁻⁴𝜒₂, and 𝑘=1𝜒₁+10⁻¹𝜒₂+10⁻⁴𝜒₃. We get 𝜙₁=0.36,𝜙₂=1.6×10⁻⁵,𝑘̃₁=0.2130,𝑐=1.0412×10⁻³.(3.9)

Example 3.5. 𝑌₂=(0.25,0.75)²,  𝑌₃=(0.2,0.8)²⧵𝑌₂,  𝜙=1𝜒₁+10⁻⁴𝜒₂, and 𝑘=1𝜒₁+10⁻¹𝜒₂+10⁻⁴𝜒₃. We get 𝜙₁=0.64,𝜙₂=2.5×10⁻⁵,𝑘̃₁=0.4519,𝑐=4.3000×10⁻³.(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 𝜀→0. In addition, the analysis in [11] suggests that it is appropriate to compare 𝜌̃_𝑖 to 𝜒_𝑖𝜌_𝜀. However, this notion of convergence does not give information about the rate at which 𝜌̃_𝑖−𝜒_𝑖𝜌_𝜀 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 𝜀→0. 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 𝑓≡0 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 Γ_𝐷={0}×[0,1]∪{1}×[0,1]. On Γ_𝐷 we define 𝜌_𝐷(0,𝑦)=1,𝜌_𝐷(1,𝑦)=0,𝑦∈[0,1]. On the lateral sides of Ω the Neumann no-flow condition is imposed. Also, let 𝜌₀≡1. Thus the flow in the micro-model goes from left to right, and the solution evolves towards the stationary solution ⟨1−𝑥,1−𝑥⟩. 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=10⁻⁴. The solution of the micro-model 𝜌 depends on the number of cells in Ω≡Ω^𝜀, 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 𝑡_𝑛 by 𝜌^𝑛𝜀,ℎ where ℎ denotes the grid parameter for the micro-model.

For the macro-model, we use 𝜌̃_𝑖,0=𝜌₀≡1. Also, we use 𝜌̃_𝐷,𝑖=𝜌_𝐷 for 𝑖=1,2 when 𝑘̃₂≢0. If 𝑘̃₂≡0, then the boundary condition for 𝜌̃₂ 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 {⟨𝜌̃^𝑛1,ℎ₁,𝜌̃^𝑛2,ℎ₂⟩}^𝑁𝑛=1 and the micro-model (2.8a) and (2.8b) for 𝜌^𝑛𝜀,ℎ. The plots of 𝜌^𝑛ℎ and 𝜌̃^𝑛𝑖,ℎ are in Figure 2, and a different view is shown in Figure 1(d).

(a)

(b)

(c)

(d)

Figure 2 

Numerical approximation at 𝑡𝑛=3×10−2 of (a)-(b) the solutions of the two components of the macro-model (4.2), and (c) the solutions of the micro-model (2.8a) and (2.8b). Figure (d) shows the side view of (a, b, c) at 𝑦=0.4 and 𝑦=0.5. Data from Example 3.1 with 1/𝜀=5 and ℎ=ℎ1=ℎ2=1/50. Note that (a) and (b) have different vertical scales.

The heterogeneous structure of Ω^𝜀 is well visible from the behavior of the micro-model solution 𝜌^𝑛ℎ. 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 𝜌̃^𝑛1,ℎ₁ which envelopes 𝜌^𝑛ℎ𝜒₁, while 𝜌̃^𝑛2,ℎ₂ envelopes 𝜌^𝑛ℎ𝜒₂ 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 𝑘̃₂≡0, 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 𝜌̃₁. The latter satisfies, however, the homogeneous boundary condition at 𝑥=1. In particular, with constant 𝜙₂, 𝑐, we have 𝜌̃₂(𝑥,𝑦,𝑡)=𝜌̃_2,0(𝑥,𝑦)𝑒^−(𝑐/𝜙_2)𝑡+^𝑡0𝜌̃₁(𝑥,𝑦,𝑠)𝑒^−(𝑐/𝜙_{2)(𝑡−𝑠)}𝑑𝑠.(3.11) Thus, for small 𝑡 and small 𝑐/𝜙₂, 𝜌̃₂(1,𝑦,𝑡) is away from 0, but, as time increases, its magnitude decreases proportionally to 𝑒^−(𝑐/𝜙_2)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.

(a)

(b)

(c)

(d)

Figure 3 

Numerical approximation at 𝑡𝑛=3×10−2 of (a) and (b) the solutions of the two components of the macro-model (4.2) and (c) the solutions of the micro-model (2.8a) and (2.8b). Figure (d) shows the side view of (a, b, c) at 𝑦=0.4 and 𝑦=0.5. Data from Example 3.2 with 1/𝜀=5 and ℎ=ℎ1=ℎ2=1/50.

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),  𝑙≥1. We use uniform mesh with ℎ=ℎ₁=ℎ₂=𝜀/20 for now. See Section 3.2.4 for ℎ_𝑖≫ℎ 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 𝜒_𝑖 are involved. While the results in [11] use two-scale convergence with 𝜀, that is, rely on special periodic test functions, we compare 𝜌^𝑛𝜀,ℎ and 𝜌̃^𝑛𝑖,ℎ directly. We see that it is easy to do so only in the connected region Ω₁.

We define a quantity 𝑒^𝑛𝜀=(𝜌^𝑛𝜀,ℎ−𝜌̃^𝑛1,ℎ₁)𝜒₁ to be used in comparison. We also define, for a fixed 𝑡∈(0,𝑇] and some function 𝑧, ‖𝑧(𝑥‖,𝑧𝑦(;𝑥𝑡,)𝑦‖;𝑡)_∗,𝐿2‖_∗,𝐿1∶=¹⁰||𝑧(𝑥,0.5;𝑡)||𝑑𝑥,∶=¹⁰||𝑧(𝑥,0.5;𝑡)||²𝑑𝑥^1/2,(3.12) which will be applied to 𝑒^𝑛𝜀. Clearly ‖⋅‖_∗,𝑊 is not a norm but is a useful quantity of interest.

In Tables 1, 2, 3, and 4 we present ‖𝑒^𝑛𝜀‖_∗,𝑊 for different 𝜀 and 𝑊=𝐿₂,𝐿₁. For each case we consider convergence of ‖𝑒^𝑛𝜀‖_∗,𝑊 with 𝜀↓0.

Table 1 

Comparison of solutions to the macro-model with those to the micro-model for different values of 𝜀 with data as in Example 3.2.

Table 2 

Comparison of the macro-model to the micro-model for different values of 𝜀 as in Example 3.3.

Table 3 

Comparison of the macro-model to the micro-model for different values of 𝜀 Example 3.4.

Table 4 

Comparison of the macro-model to the micro-model for different values of 𝜀 Example 3.5.

From homogenization theory it is not clear what order of convergence should be expected. We observe that ‖𝑒^𝑛𝜀‖_∗,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 𝜌̃₁ and 𝜌̃₂ 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, 𝜙₁ and 𝑘̃₁ 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 ℎ=ℎ₁=ℎ₂. 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 ℎ≪ℎ₁ in the macro-model.

Example 3.6. We use data from Example 3.2 with 𝜀=0.2. Consider a fixed microgrid with ℎ⁻¹=100 and a few cases of macrogrid with ℎ¹⁻¹=ℎ²⁻¹=10,50,100. Now we compare the micro- and macrosolutions 𝜌^𝑛𝜀,ℎ and 𝜌_1,ℎ1, see Table 5. We also compare the run time of the macro-model to that of the reference micro-model which is 120 seconds.

Table 5 

Comparison of the micro-model to the macro-model as in Example 3.6.

We see that, as predicted above, the macro-model run time at ℎ_𝑖=ℎ 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 ℎ_𝑖≫ℎ, 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 ℎ_𝑖 is by a factor of 10 coarser than ℎ 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 ℎ₁≠ℎ₂ are implemented, see Section 4.

Influence of Parameter Grid
Last we address the effect of the computed parameters 𝑐, 𝑘̃₁ 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 ℎ⁻¹=60. Use ℎ¹⁻¹=ℎ²⁻¹=60. Compute numerical parameters 𝑐_ℎ𝑐, 𝑘̃_1,ℎ𝑘1 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.

Table 6 

Sensitivity of the macro-model solution to the grid used to compute 𝑐_ℎ𝑐≈𝑐 and 𝑘̃_1,ℎ𝑘1≈𝑘̃₁ as Example 3.7.

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 𝑘̃₁≫𝑘̃₂, that is, that the first component varies in space faster than the second. As a special case, this includes the case 𝑘̃₂=0. We consider two triangulations 𝒯_ℎ𝑖, 𝑖=1,2, which will be used for 𝜌̃_ℎ𝑖, respectively. In principle, these can be chosen independently. In fact, we allow the triangulations to vary in time and thus consider 𝒯^𝑛ℎ_𝑖 and the associated spaces 𝑉^𝑛ℎ_𝑖 as in (2.7). To avoid the loss of accuracy due to excessive interpolation and intergrid projections and because 𝑘̃₁≫𝑘̃₂, we assume that 𝒯^𝑛ℎ₁ is a refinement of the partition 𝒯^𝑛ℎ₂.

We need intergrid operators to handle two components that live on separate grids. Let Π∶𝑉^𝑛ℎ₂→𝑉^𝑛ℎ₁ be the interpolation operator and Π′∶𝑉^𝑛ℎ₁→𝑉^𝑛ℎ₂ the 𝐿₂ projection defined by Π^𝜙_ℎ1^,𝜓_ℎ2∶=𝜙^ℎ₁,Π_𝜓ℎ2,∀𝜓_ℎ2∈𝑉_ℎ2.(4.1)

Now we define the discrete solutions. At each time step 1≤𝑛≤𝑁, we seek 𝜌̃^𝑛𝑖,ℎ_𝑖∈𝑉^𝑛ℎ_𝑖, 𝑖=1,2, which satisfy the discrete problem, that is, (2.4) restricted to the finite dimensional subspaces 𝑉^𝑛ℎ_𝑖 so that, for all 𝜓_ℎ1∈𝑉^𝑛ℎ₁, for all 𝜉_ℎ2∈𝑉^𝑛ℎ₂𝜙¹𝜌̃^𝑛1,ℎ₁−𝜌̃^{𝑛−11,ℎ}₁Δ𝑡_𝑛,𝜓_ℎ1+𝑘̃¹𝜙₂∇𝜌̃^𝑛1,ℎ₁,∇𝜓_ℎ1+𝑐𝜌̃^𝑛1,ℎ₁−Π𝜌̃^𝑛2,ℎ₂,𝜓^ℎ₁=𝑓,𝜓^ℎ₁,𝜌̃^𝑛2,ℎ₂−𝜌̃^{𝑛−12,ℎ}₂Δ𝑡_𝑛,𝜉_ℎ2+𝑘̃²𝜌̃_01,ℎ1∇𝜌̃^𝑛2,ℎ₂,∇𝜉_ℎ2+𝑐𝜌̃^𝑛2,ℎ₂−Π^𝜌̃^𝑛1,ℎ₁,𝜉^ℎ₂=0,,𝜌̃^02,ℎ₂=ℐ^ℎ₁𝜌̃_1,0,ℐ_ℎ2𝜌̃_2,0inΩ.(4.2) If 𝑘̃₂=0, the system (4.2) is modified appropriately so that 𝜌̃₂(⋅,𝑡)∈𝐿²(Ω) instead of 𝐻^1𝐷(Ω), and we do not impose boundary conditions on this component. In fact, instead of (2.7) we define 𝑉^𝑛ℎ₂=𝑣_ℎ∈𝐶Ω∶∀𝑇∈𝒯^𝑛ℎ,𝑣_ℎ∣_𝑇∈𝒫_𝑘(𝑇).(4.3) (It holds that 𝑣_ℎ∈𝐶(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 1≤𝑛≤𝑁 and 𝑖=1,2. We denote 𝑒_𝑖,ℎ𝑖(𝑡_𝑛)=𝜌̃^𝑛𝑖,ℎ_𝑖−𝜌̃^𝑛𝑖 and define the energy norm for the product space ||‖⟨𝑧,𝑤⟩‖||²𝑡_𝑛∶=𝜙₁‖‖𝑧𝑡_𝑛‖‖²+𝜙₂‖‖𝑤𝑡_𝑛‖‖²+^𝑛_𝑚₌₁Δ𝑡_𝑚𝑘̃¹||𝑧𝑡_𝑚||²¹+𝑘̃₂||𝑤𝑡_𝑚||²¹.(4.4)

Assuming that ⟨𝜌̃₁,𝜌̃₂⟩ is sufficiently smooth we have the following a-priori estimate proven in the appendix.

Theorem 4.1. Let ⟨𝜌̃₁,𝜌̃₂⟩,{⟨𝜌̃^𝑛1,ℎ₁,𝜌̃^𝑛2,ℎ₂⟩}^𝑁𝑛=1 be the smooth solutions of (2.4) and (4.2), respectively. Then one has ||‖‖𝑒_1,ℎ1,𝑒_2,ℎ2‖‖||𝑡_𝑛≤𝑘̃^11/2𝐶₁ℎ₁+𝑘̃^21/2𝐶₂ℎ₂+𝐶₃max_{𝑚=1∶𝑛}Δ𝑡_𝑚+higherorderterms(h.o.t.),(4.5) where 𝐶₁, 𝐶₂, and 𝐶₃ are independent of ℎ.
Furthermore, if 𝑘̃₂=0, then ||‖‖𝑒_1,ℎ1,𝑒_2,ℎ2‖‖||𝑡_𝑛≤𝑘̃^11/2𝐶₁ℎ₁+𝐶₂ℎ²²+𝐶₃max_{𝑚=1∶𝑛}Δ𝑡_𝑚+higherorderterms(ℎ.𝑜.𝑡.).(4.6)

Now we notice that (4.5) suggests the following choice of grid parameters ℎ₁, ℎ₂. If 𝑘̃₁≫𝑘̃₂, then to balance the components of the error one can use ℎ₁≪ℎ₂. Similarly, if 𝑘̃₂=0, then (4.6) implies that ℎ₂ can be chosen to be of the order of ℎ^11/2. The use of coarse grid ℎ₂≫ℎ₁ 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 𝜂_𝑛=(∑^𝑛𝑚=1[Δ𝑡_𝑚𝑆^2𝑚+𝑇^2𝑚])^1/2 for (4.2) is composed of the temporal part 𝑇_𝑛 which adapts the time stepping and the spatial part 𝑆_𝑛 which guides the spatial grid adaptivity. We define 𝑇^2𝑛∶=Δ𝑡^𝑛3‖‖𝑘̃^11/2𝜌̃^𝑛1,ℎ₁−𝜌̃^{𝑛−11,ℎ}₁‖‖²+‖‖𝑘̃^21/2𝜌̃^𝑛2,ℎ₂−𝜌̃^{𝑛−12,ℎ}₂‖‖²+Δ𝑡_𝑛3‖‖𝑐^1/2𝜌̃^𝑛1,ℎ₁−𝜌̃^{𝑛−11,ℎ}₁−𝑐^1/2𝜌̃^𝑛2,ℎ₂−𝜌̃^{𝑛−12,ℎ}₂‖‖²,(4.7) and 𝑆^2𝑛=𝑆_1,𝑛+𝑆_2,𝑛, where 𝑆_𝑖,𝑛=∑_{𝑇𝑖∈𝒯ℎ𝑖}𝑆^{𝑛𝑖,𝑇}_𝑖, and where, as usual (see [23]), one defines 𝑆^{𝑛𝑖,𝑇}_𝑖∶=𝜃^2𝑖,𝑇_𝑖‖‖𝑅^{𝑛𝑖,𝑇}_𝑖‖‖²+12_{𝐸∈ℰ𝑇𝑖}𝛾^2𝑖,𝐸‖‖𝑅^{𝑛𝑖,𝐸}‖‖².(4.8) The element and edge residuals in 𝑆^{𝑛𝑖,𝑇}_𝑖,𝑖=1,2, are given by 𝑅^{𝑛𝑖,𝑇}_𝑖=𝑓_𝑖−𝜙_𝑖𝜌̃^𝑛𝑖,ℎ_𝑖−𝜌̃^{𝑛−1𝑖,ℎ}_𝑖Δ𝑡_𝑛𝑅+∇⋅𝑘̃_𝑖∇𝜌̃^𝑛𝑖,ℎ_𝑖+(−1)^𝑖𝑐𝜌̃^𝑛1,ℎ₁−𝜌̃^𝑛2,ℎ₂,(4.9)_{𝑛𝑖,𝐸}=⎧⎪⎨⎪⎩𝑘̃^𝑖−𝑘̃𝜕_𝜈𝜌̃^𝑛𝑖,ℎ_𝑖_𝐸if𝐸⊄𝜕Ω,_𝑖0,𝜕_𝜈𝜌̃^𝑛𝑖,ℎ_𝑖oifth𝐸er⊂wiΓse_𝑁.,(4.10) Here [𝑧]_𝐸 denotes the jump of the flux 𝑧 across the edge 𝐸 of an element. Usually 𝑓₁≡𝑓, 𝑓₂≡0. Also, if 𝑘̃₂=0, then 𝑅^𝑛2,𝐸≡0.

The scaling constants 𝜃_𝑖,𝑇,𝛾_𝑖,𝐸 take into account the contribution of the exchange term 𝑐(𝜌̃₁−𝜌̃₂) and are defined so that the estimators remain robust when the coefficients of the problem change substantially: 𝜃𝜃_1,𝑇=minℎ_𝑇1𝑘̃^1−1/2,max𝑐^−1/2,ℎ_𝑇1𝑘̃^2−1/2,𝑇¹∈𝒯^𝑛ℎ₁∪ℰ^𝑛ℎ₁,(4.11)_2,𝑇=minℎ_𝑇2𝑘̃𝛾^2−1/2,max𝑐^−1/2,ℎ_𝑇2𝑘̃^1−1/2,𝑇²∈𝒯^𝑛ℎ₂∪ℰ^𝑛ℎ₂,(4.12)_𝑖,𝐸=2ℎ^𝐸−1/2𝜃_𝑖,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 𝑘̃₂≡0 when 𝜌̃₂∈𝐿²(Ω). We have the following result on reliability of the estimator proven in the appendix.

Theorem 4.2. Let ⟨𝜌̃₁,𝜌̃₂⟩, {⟨𝜌̃^𝑛1,ℎ₁,𝜌̃^𝑛2,ℎ₂⟩}^𝑁𝑛=1 be the solution of (1.2a) and (1.2b), (2.3) and (4.2), respectively. Then ||‖‖𝑒_1,ℎ1,𝑒_2,ℎ2‖‖||𝑡_𝑛≤𝜂_𝑛+h.o.t.;∀𝑛≥1(4.14)

4.3. Adaptive Two-Grid Discretization and Implementation

Consider a fixed time step 𝑛=𝑛_𝑜 for which some triangulation is chosen, the solution is found, and the spatial error indicators 𝑆_𝑖,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 𝑆_1,𝑛𝑜>3𝑆_2,𝑛𝑜, refine only 𝒯^𝑛𝑜_ℎ1. If 𝑆_2,𝑛𝑜>3𝑆_1,𝑛𝑜, refine only 𝒯^𝑛𝑜_ℎ2. Otherwise, refine both. (2)In 𝒯_ℎ𝑖, 𝑖=1,2 selected in Step 1, refine any 𝑇∈𝒯_ℎ,𝑖 for which 𝑆^𝑛𝑜_𝑖,𝑇≥0.5max_{𝑇∈𝒯ℎ𝑖}𝑆^𝑛𝑜_𝑖,𝑇. (3)Enforce the requirement that 𝒯^𝑛ℎ₁ is a refinement of 𝒯^𝑛ℎ₂ by adding extra elements as needed.
The steps in 𝒜 should be repeated a certain number 𝑁_𝒜 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=10⁻⁴. Other data is as in Section 3.2.1.
Consider a fixed 𝑛₀=200 with a uniform triangulation 𝒯^𝑛𝑜_ℎ1=𝒯^𝑛𝑜_ℎ2 with ℎ₁=ℎ₂=0.1. Then apply 𝑁_𝐴=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 𝑡_𝑛0 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 𝑘̃₂=0. This can be compared to the a-priori estimate in Theorem 4.1 from which we know the convergence is 𝑂(ℎ₁+ℎ²²+Δ𝑡_𝑛).
In Table 7 we provide the details on 𝒯^𝑛𝑜_ℎ𝑖 and compare the effectiveness of the local two-grid refinement by algorithm 𝒜 with uniform grid refinement. With adaptive refinement we get 𝑆_𝑛𝑜=0.35834 with 435+121=556 unknowns, while the uniform refinement needs 441+441=882 unknowns to achieve comparable 𝑆_𝑛𝑜=0.39221.

Table 7 

Adaptive versus uniform refinement.

(a)

(b)

(c)

(d)

Figure 4 

Sixth level refinement iteration at 𝑡𝑛𝑜=2×10−2. Figures (a) and (b): plots of 𝜌̃𝑛𝑜1,ℎ1 and 𝒯𝑛𝑜ℎ1. Figures (c), (e), and (d): plots of 𝜌̃𝑛𝑜2,ℎ2 and 𝒯𝑛𝑜ℎ2.

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

Implementation
The decision to use two grids implies that we have to interpolate between the finite element spaces 𝑉^𝑛ℎ₁ and 𝑉^𝑛ℎ₂, see (4.1). Suppose that {𝜁_𝑖,𝑗}_𝑗,  𝑖=1,2, is a basis for 𝑉^𝑛ℎ_𝑖. Then the operator Π in (4.1) is represented by the matrix 𝕀_ℎ1,ℎ2 whose components are given by 𝕀_ℎ1,ℎ2(𝑗,𝑘)=(𝜁_1,𝑗,𝜁_2,𝑘).
Consider now and compare the computational cost associated with one-grid and two-grid approaches. To solve (4.2) with one-grid 𝑉^𝑛ℎ₁ we need to assemble the stiffness matrix 𝑆_1,ℎ1(𝑗,𝑘)=(𝑘̃₁∇𝜁_1,𝑗,∇𝜁_1,𝑘), the mass matrix 𝑀_1,ℎ1(𝑗,𝑘)=(𝜁_1,𝑗,𝜁_1,𝑘), and the block matrix 𝐵_one-grid=⎡⎢⎢⎣−Δ𝑡𝑆^1,ℎ1+𝜙₁+𝑐Δ𝑡𝑀_1,ℎ1−𝑐Δ𝑡𝑀_1,ℎ1−𝑐Δ𝑡𝑀_1,ℎ1𝜙²+𝑐Δ𝑡𝑀_1,ℎ1⎤⎥⎥⎦.(4.15) At each time step we solve the linear system with 𝐵_one−grid; this is referred to as 𝕊_one-grid.
To solve (4.2) with two grids and two spaces 𝑉^𝑛ℎ₁,𝑉^𝑛ℎ₂, we need to assemble the stiffness matrix 𝑆_1,ℎ1(𝑗,𝑘)=(𝑘̃₁∇𝜁_1,𝑗,∇𝜁_1,𝑘), the mass matrices 𝑀_1,ℎ1(𝑗,𝑘)=(𝜁_1,𝑗,𝜁_1,𝑘), 𝑀_2,ℎ2(𝑗,𝑘)=(𝜁_2,𝑗,𝜁_2,𝑘), the interpolation matrix 𝕀_ℎ1,ℎ2(𝑗,𝑘)=(𝜁_1,𝑗,𝜁_2,𝑘), and the block matrix 𝐵_two-grid=⎡⎢⎢⎣−Δ𝑡𝑆^1,ℎ1+𝜙₁+𝑐Δ𝑡𝑀_1,ℎ1−𝑐Δ𝑡𝕀^𝑇ℎ_1,ℎ2−𝑐Δ𝑡𝕀_ℎ1,ℎ2𝜙²+𝑐Δ𝑡𝑀_2,ℎ2⎤⎥⎥⎦.(4.16) To finish we solve the linear system which we refer to as 𝕊_two-grid.

Suppose now we solve (4.2) and wish to maintain 𝑆_𝑛≤𝛿 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 ℎ₁=ℎ₂=1/80. Using the two-grid approach we only need ℎ₁=1/80,ℎ₂=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.

Table 8 

Cost of one-grid versus two-grid discretization in a simulation with 1,000 time steps.

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.

Appendix

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 𝒯^𝑛ℎ₁ is a refinement of 𝒯^𝑛ℎ₂, we can rewrite (4.2) as 𝜙¹𝜌̃^𝑛1,ℎ₁−𝜌̃^{𝑛−11,ℎ}₁Δ𝑡_𝑛,𝜓_ℎ1+𝑘̃¹𝜙₂∇𝜌̃^𝑛1,ℎ₁,∇𝜓_ℎ1+𝑐𝜌̃^𝑛1,ℎ₁−𝜌̃^𝑛2,ℎ₂,𝜓^ℎ₁=𝑓,𝜓^ℎ₁,𝜌̃^𝑛2,ℎ₂−𝜌̃^{𝑛−12,ℎ}₂Δ𝑡_𝑛,𝜉_ℎ2+𝑘̃²∇𝜌̃^𝑛2,ℎ₂,∇𝜉_ℎ2+𝑐𝜌̃^𝑛2,ℎ₂−𝜌̃^𝑛1,ℎ₁,𝜉^ℎ₂=0.(A.1) Next, for 𝑘̃₂≢0, as in [26] we define the elliptic projection ⟨𝑃𝜌̃₁,𝑃𝜌̃₂⟩ of ⟨𝜌̃₁,𝜌̃₂⟩ into 𝑉_ℎ1×𝑉_ℎ2 via ∇𝑃𝜌̃_𝑖−𝜌̃_𝑖,∇𝜒=0,∀𝜒∈𝑉_ℎ𝑖,𝑖=1,2.(A.2) Applying (A.2) for 𝑖=1,2 into the weak formulation (2.4), with arbitrary 𝜓,𝜉∈𝐻^1𝐷(Ω) we arrive at 𝜙¹𝜕𝑃𝜌̃₁𝜕𝑡,𝜓+𝑘̃¹∇𝑃𝜌̃₁,∇𝜓+𝑐𝜌̃₁−𝜌̃₂,𝜓=(𝑓,𝜓)+𝜙₁𝜕𝑃𝜌̃₁−𝜌̃₁𝜙𝜕𝑡,𝜓₂,𝜕𝑃𝜌̃₂𝜕𝑡,𝜉+𝑘̃²∇𝑃𝜌̃₂,∇𝜉+𝑐𝜌̃₂−𝜌̃₁,𝜉=𝜙₂𝜕𝑃𝜌̃₂−𝜌̃₂𝜕𝑡,𝜉.(A.3)
Next we define 𝜕^𝑛𝑤=(𝑤(𝑡_𝑛)−𝑤(𝑡_𝑛−1))/Δ𝑡_𝑛, Δ_𝑛(𝑤)=𝜕^𝑛𝑤−(𝜕𝑤/𝜕𝑡)(𝑡_𝑛), subtract (A.3) from (A.1), add the resulting equations, and use as test functions 𝜓=𝜌̃^𝑛1,ℎ₁−𝑃𝜌̃^𝑛1 and 𝜉=𝜌̃^𝑛2,ℎ₂−𝑃𝜌̃^𝑛2 to get 𝜙₁‖‖𝜌̃^𝑛1,ℎ₁−𝑃𝜌̃^𝑛1‖‖²+𝜙₂‖‖𝜌̃^𝑛2,ℎ₂−𝑃𝜌̃^𝑛2‖‖²+𝑘̃₁Δ𝑡_𝑛|||𝜌̃_𝑛1,ℎ1−𝑃𝜌̃^𝑛1|||²¹+𝑘̃₂Δ𝑡_𝑛|||𝜌̃_𝑛2,ℎ2−𝑃𝜌̃^𝑛2|||²¹≤+𝜙₁Δ𝑡𝜌_𝑛̃^{𝑛−11,ℎ}₁−𝑃𝜌̃^1𝑛−1,𝜌̃^𝑛1,ℎ₁−𝑃𝜌̃^𝑛1+𝜙²𝜌̃^{𝑛−12,ℎ}₂−𝑃𝜌̃^2𝑛−1,𝜌̃^𝑛2,ℎ₂−𝑃𝜌̃^𝑛2𝜙¹𝜕𝑃𝜌̃₁−𝜌̃₁𝜕𝑡,𝜌̃^𝑛1,ℎ1−𝑃𝜌̃^𝑛1+Δ𝑡^𝑛𝜙²𝜕𝑃𝜌̃₂−𝜌̃₂𝜕𝑡,𝜌̃^𝑛2,ℎ2−𝑃𝜌̃^𝑛2−Δ𝑡_𝑛𝜙¹Δ_𝑛𝑃𝜌̃₁,𝜌̃^𝑛1,ℎ₁−𝑃𝜌̃^𝑛1−Δ𝑡^𝑛𝜙²Δ_𝑛𝑃𝜌̃₂,𝜌̃^𝑛2,ℎ₂−𝑃𝜌̃^𝑛2.(A.4) With the help of standard inequalities, summing the equations from 𝑛=1…,𝑁, and applying the discrete Gronwall's lemma with 𝛽=2max{𝜙₁,𝜙₂}, we get |||‖‖𝜌̃_𝑛1,ℎ1−𝑃𝜌̃^𝑛1,𝜌̃^𝑛2,ℎ₂−𝑃𝜌̃^𝑛2‖‖|||²≤𝑒^𝛽𝑡𝑁𝜙¹‖‖𝜌̃^01,ℎ₁−𝑃𝜌̃⁰¹‖‖²+𝜙₂‖‖𝜌̃^02,ℎ₂−𝑃𝜌̃⁰²‖‖²+^𝑁_𝑛=1Δ𝑡_𝑛𝜙₁‖‖‖‖𝜕𝑃𝜌̃¹−𝜌̃₁𝜕𝑡‖‖‖‖²+‖‖Δ_𝑛𝑃𝜌̃₁‖‖²+_𝑁_𝑛=1Δ𝑡_𝑛𝜙₂‖‖‖𝜕(𝑃𝜌̃²−𝜌̃₂)𝜕𝑡‖‖‖²+‖‖Δ_𝑛(𝑃𝜌̃₂)‖‖².(A.5) Next step is to estimate the quantities ‖𝜕(𝑃𝜌̃_𝑖−𝜌̃_𝑖)/𝜕𝑡‖,‖Δ_𝑛(𝑃𝜌̃_𝑖)‖, 𝑖=1,2. This is done using standard properties of elliptic projections as in [25, Lemmas  13.2 and 13.4, pages 233 and 241] assuming 𝜌̃_𝑖,𝜕𝜌̃_𝑖/𝜕𝑡 are smooth enough. Combining these we arrive at the desired inequality (4.5).
For the case 𝑘̃₂≡0, we define 𝑃^∗𝜌̃₂ as the 𝐿²-projection of 𝜌̃₂ over 𝑉^𝑛ℎ₂ via 𝑃^∗𝜌̃₂−𝜌̃₂,𝜒=0∀𝜒∈𝑉^𝑛ℎ₂.(A.6) The proof follows along the same lines as for 𝑘̃₂≢0 except that we do not (and cannot) use any terms with ∇𝜌̃₂.

B. Proof of Theorem 4.2

Proof. The proof is a combination of techniques in [13–15] which we extend to a coupled system and propose an estimator robust with respect to the five parameters of the problem.
First we define the semidiscrete problem: find {⟨𝜌̃^𝑛1,𝜌̃^𝑛2⟩}^𝑁𝑛=1∈(𝐻^1𝐷(Ω))² so that for all ⟨𝜓,𝜉⟩∈(𝐻^1𝐷(Ω))²𝜙¹𝜌̃^𝑛1,𝜙𝜓+𝑘̃₁∇𝜌̃^𝑛1,∇𝜓+𝑐𝜌̃^𝑛1−𝜌̃^𝑛2,𝜓=(𝑓,𝜓)+𝜙₁𝜌̃^1𝑛−1,𝜓₂,𝜌̃^𝑛2,𝜉+𝑘̃₂∇𝜌̃^𝑛2,∇𝜉+𝑐𝜌̃^𝑛2−𝜌̃^𝑛1,𝜉=𝜙₂𝜌̃^2𝑛−1,𝜉.(B.1) We estimate the error between the solution of (2.4) and (B.1) and this is how the time estimator 𝑇^𝑛 (4.7) arises; see [15] for details.
Now the semidiscrete system (B.1) is a stationary coupled reaction-diffusion model of a type we considered in [13] and for which we proposed a spatial a posteriori error estimator accounting for the coupling terms in the system. That estimator is robust, that is, the efficiency ratio remains essentially constant when the coefficients change by orders of magnitude; this is relevant for our problem since we may have small 𝑘̃₂. The robustness in [13] is achieved by an appropriate scaling of constants in the estimator; the scaling in (4.11)–(4.13) serves the same purpose.
Recall the standard set-up first. Fix 𝑖=1,2, and for any 𝑇∈𝒯^𝑛ℎ_𝑖 denote by 𝜔_𝑇 the set of all elements in 𝒯^𝑛ℎ_𝑖 that share at least one vertex with 𝑇. For the quasi-interpolator 𝑄_ℎ defined in [14, Lemma  3.1, page 482], we have that for any 𝑣∈𝐻^𝑘(𝑤_𝑇), 0≤𝑘≤1, ‖‖∇^𝑙𝑣−𝑄_ℎ𝑣‖‖_𝑇≤𝐶ℎ^{𝑇𝑘−𝑙}‖‖∇^𝑘𝑣‖‖𝜔_𝑇0≤𝑙≤𝑘≤1,(B.2) where the constant 𝐶 is independent of ℎ, 𝑣, and where ∇⁰𝑣=𝑣. The plan is to extend (B.2) for 𝑙=0 to an inequality involving the form ℬ_𝑇=𝜔_𝑇𝑐(𝜉−𝜓)²+𝑘̃₂(∇𝜓)²+𝑘̃₁(∇𝜉)²^1/2,(B.3) which involves the coupling term as well as other coefficients of the problem and thus leads to robustness.
If 𝑘̃₂>0, let 𝜉,𝜓∈𝐻¹(Ω) and recall that 𝑉^𝑛ℎ₂⊆𝑉^𝑛ℎ₁. If 𝑘̃₂≡0, we consider 𝜉∈𝐻^1𝐷(Ω), 𝜓∈𝐿²(Ω), and away from the boundary on some Ω^∘⊊Ω we have 𝑉^𝑛ℎ₂(Ω^∘)⊆𝑉^𝑛ℎ₁(Ω^∘). For a fixed 𝑛, 𝑖, let 𝜉_ℎ𝑖, 𝜓_ℎ𝑖 be the quasi-interpolators of 𝜙, 𝜓 in 𝑉^𝑛ℎ_𝑖, respectively. Note that 𝜓_ℎ2 is defined correctly when 𝑘₂=0 and 𝜓∈𝐿²(Ω). To get the desired estimate, we apply (B.2) with 𝑇∈𝒯^𝑛ℎ₁, and 𝑙=0, 𝑘=1, to get ‖𝜉−𝜉_ℎ1‖_𝑇≤ℎ_𝑇‖∇𝜉‖𝜔_𝑇𝜉‖𝜔_𝑇≤(ℎ_𝑇/𝑘̃^11/2)ℬ_𝑇. To extend, we add and subtract 𝜓−𝜓_ℎ1 and use the triangle inequality followed by (B.2) with 𝑘=𝑙=0 and 𝑘=1,𝑙=0 to get ‖‖𝜉−𝜉_ℎ1‖‖_𝑇≤‖‖𝜉−𝜉_ℎ1−𝜓−𝜓_ℎ1‖‖_𝑇+‖‖𝜓−𝜓_ℎ1‖‖_𝑇≤‖𝜉−𝜓‖𝜔_𝑇+ℎ_𝑇‖∇𝜓‖𝜔_𝑇≤√2max1𝑐^1/2,ℎ^𝑇𝑘̃^21/2ℬ^𝑇.(B.4) Thus we have by (4.11) ‖‖𝜉−𝜉_ℎ1‖‖_𝑇≤𝜃_1,𝑇ℬ_𝑇,(B.5) which is the desired element estimate.
To prove the edge interpolation estimate for an edge 𝐸 of the element 𝑇, we use the following trace inequality [43, Lemma  3.1, page 645] for 𝑣∈𝐻¹(𝑇): ‖𝑣‖_𝐸≤𝐶ℎ^𝑇−1/2‖𝑣‖_𝑇+‖𝑣‖^𝑇1/2‖∇𝑣‖^𝑇1/2,(B.6) where 𝐶 is a constant independent of 𝑣, ℎ_𝑇. Let 𝑣=𝜉−𝜉_ℎ1. By (B.2) ‖‖∇𝜉−𝜉_ℎ1‖‖_𝑇≤‖∇𝜉‖𝜔_𝑇≤ℎ^𝑇−1𝜃_1,𝑇ℬ_𝑇.(B.7) Applying (B.6) followed by (B.5) and (B.7) we get ‖‖𝜉−𝜉_ℎ1‖‖_𝐸≤ℎ^𝑇−1/2‖‖𝜉−𝜉_ℎ1‖‖_𝑇+‖‖𝜉−𝜉_ℎ1‖‖^𝑇1/2‖‖∇𝜉−𝜉_ℎ1‖‖^1/2≤ℎ^𝑇−1/2𝜃_1,𝑇ℬ_𝑇+𝜃^1/21,𝑇ℬ^𝑇1/2ℎ^𝑇−1/2𝜃^1/21,𝑇ℬ^𝑇1/2.(B.8) Thus we obtain ‖𝜉−𝜉_ℎ1‖_𝐸≤𝛾_1,𝐸ℬ_𝑇, and the estimate for the second component follows similarly if 𝑘̃₂≢0.
When 𝑘̃₂≡0, we have 𝜉∈𝐻^1𝐷(Ω),𝜓∈𝐿²(Ω), and (4.12) is reduced to 𝜃_2,𝑇=max{1/𝑐^1/2,ℎ_𝑇/𝑘̃^11/2}. Also, 𝛾_2,𝑇 does not need to be defined since 𝑅^𝑛2,𝐸≡0. Now we add and subtract 𝜓−𝜓_ℎ1 and use the triangle inequality followed by (B.2) with 𝑘=𝑙=0 and 𝑘=1, 𝑙=0 to get ‖‖𝜓−𝜓_ℎ2‖‖_𝑇≤‖‖𝜉−𝜉_ℎ1−𝜓−𝜓_ℎ1‖‖_𝑇+‖‖𝜉−𝜉_ℎ1‖‖_𝑇≤‖𝜉−𝜓‖𝜔_𝑇+ℎ_𝑇‖∇𝜉‖𝜔_𝑇≤√2max1𝑐^1/2,ℎ^𝑇𝑘̃^11/2ℬ^𝑇.(B.9) Thus, by (4.12) we have ‖𝜓−𝜓_ℎ1‖_𝑇≤𝜃_2,𝑇ℬ_𝑇, which is the desired edge estimate.
Combining these interpolation estimates with the usual technique of upper bounds for residual a posteriori error estimators extended to systems in [13] completes the proof the theorem.

Acknowledgments

The authors would like to thank Professor Ralph Showalter from Oregon State University for numerous discussions on the double-diffusion models. they would like also to acknowledge the package 𝑖FEM [44] which was used to refine and coarsen the grid in our adaptive implementation. they are also grateful to the anonymous referee whose remarks helped to improve the paper. This work was partially supported by the Grants NSF DMS-0511190, DMS-1115827, DOE 98089, and by FAPESP/Brazil.