Abstract
We consider discretizations for reaction–diffusion systems with nonlinear diffusion in two space dimensions. The applied model allows to handle heterogeneous materials and uses the chemical potentials of the involved species as primary variables. We propose an implicit Voronoi finite volume discretization on arbitrary, even anisotropic, Voronoi meshes that allows to prove uniform, mesh-independent global upper and lower bounds for the chemical potentials. These bounds provide one of the main steps for a convergence analysis for the fully discretized nonlinear evolution problem. The fundamental ideas are energy estimates, a discrete Moser iteration and the use of discrete Gagliardo–Nirenberg inequalities.
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1 Introduction and model equations
In a bounded domain \(\varOmega \subset \mathbb {R}^2\) we consider \(m\) species \(X_\nu \) with initial densities \(U_\nu \) which underly diffusion processes and undergo chemical reactions. The relation between the densities \(u_\nu \) of the species \(X_\nu \) and the corresponding chemical potentials \(v_\nu \) is assumed to be given by Boltzmann statistics, i.e.,
The reference densities \(\overline{u}_\nu \) may depend on the spatial position and express the possible heterogeneity of the system under consideration. For the mass fluxes \(j_\nu \) we make the ansatz
with diffusion coefficients \(D_{\nu }:\varOmega \times \mathbb {R}^{m}\rightarrow \mathbb {R}_+\) which are allowed to depend on the space variable and the state variable. To describe chemical reactions we introduce a finite subset \(\mathcal {R}\subset \mathbb {Z}_+^m\times \mathbb {Z}_+^m\). Each pair \((\varvec{\alpha },\varvec{\beta })\in \mathcal {R}\) represents the vectors of stoichiometric coefficients of a reversible reaction, written in the form
According to the mass action law, the net rate of this pair of reactions is of the form \(k_{(\varvec{\alpha },\varvec{\beta })}(a^{\varvec{\alpha }}-a^{\varvec{\beta }})\), where \(k_{(\varvec{\alpha },\varvec{\beta })}:\varOmega \times \mathbb {R}^{m}\rightarrow \mathbb {R}_+\) is a reaction coefficient, \(a_\nu :={{\mathrm{e}}}^{v_\nu }\) corresponds to the chemical activity of \(X_\nu \), and \(a^{\varvec{\alpha }}:=\prod _{\nu =1}^ma_\nu ^{\alpha _\nu }\). The net production rate of species \(X_\nu \) corresponding to all accruing reactions is
In this notation our reaction–diffusion system consists of \(m\) continuity equations with no flux boundary conditions on \(\varGamma =\partial \varOmega \):
This continuous (non-discretized) reaction–diffusion system with reversible reactions of mass action type in a formulation for heterostructures was intensively studied. In Sect. 2 we give an overview on its analytical properties, related results and corresponding references.
The aim of the paper consists in a study of a discretization scheme (backward Euler in time and Voronoi finite volume meshes in space) for problem (4). It is strongly desired to retain the analytic properties of the continuous problem also in the discretization scheme. In [24] results on invariants, discrete stationary states, and energy estimates, as well as the dissipativity of the scheme are provided. Especially, the exponential decay of the discrete free energy to its equilibrium value (independently of the underlying mesh) for classes of Voronoi finite volume meshes is obtained.
In the present paper, we prove the solvability of the discretized problems. Exploiting the results of [24], we prove as main results of the paper global with respect to time upper bounds and strictly positive lower bounds for the discretized densities. Our special aim is to find uniform bounds (being independent of the underlying Voronoi mesh). Especially anisotropic grids can be handled.
Motivated by the continuous setting we use a Moser iteration technique [38] to obtain the uniform upper and lower bounds, see [19, 29] for the continuous situation. This procedure in the discrete case involves the application of a discrete Gagliardo–Nirenberg inequality in the setting of Voronoi finite volume schemes, where the constants are uniform. Additionally, a lot of extra terms resulting from the discretization in time and space have to be managed and uniformly estimated in terms of the class of discretizations. To our knowledge, such uniform global \(L^\infty \) estimates based on Moser iteration for discretized nonlinear evolutionary systems of this kind are new. We remark that our proof is similar to finite volume techniques for proving discrete maximum principles, where special discrete functions are applied to obtain the desired result [14, 18]. But instead of testing with the positive part of the (appropriately shifted) discrete solution, as needed for the maximum principle, in the Moser iteration case we test either with a sequence of powers of the positive part of the shifted chemical activity or a sequence of powers of the negative part of the chemical potential divided by the chemical activity.
In Sect. 2 we collect the general assumptions concerning the data of the continuous problem and give a summary on results obtained for the continuous problem. The main results of the paper concerning the discretization scheme are formulated and proven in Sect. 3. We start with the description of the discretization, give a local existence result, summarize physically motivated estimates and show uniform global upper bounds of the discretized solution. Additionally, based on the asymptotic behavior of the discretized solutions, we derive uniform, global, positive lower bounds of the densities. These bounds provide one of the main steps for a convergence analysis for the fully discretized nonlinear evolution problem in the spirit of [14]. In Sect. 4 we present a numerical example illustrating the properties of the scheme.
In Appendix A we introduce a discrete Gagliardo–Nirenberg and Sobolev–Poincaré inequality proven in [1]. We derive an extended version of a discrete Gagliardo–Nirenberg inequality needed in Sect. 3.4. Appendix B contains technical lemmas necessary for the treatment of the test functions in the a priori estimates.
2 The continuous problem
2.1 General assumptions on the data
In this section we formulate basic assumptions with respect to the data of the problem, see [19, 24]. We introduce the stoichiometric subspace \(\mathcal {S}:= {{{\mathrm{span}}}}\{\varvec{\alpha }-\varvec{\beta }:(\varvec{\alpha },\varvec{\beta })\in \mathcal {R}\}\).
Definition 1
(Reaction order, see [19]) A source term of a reaction is of order \(k\), iff there exists a \(c>0\) such that
We will study the problem under the following assumptions:
-
(A1)
\(\varOmega \subset \mathbb {R}^2\) is a bounded polygonal domain, \(\varGamma :=\partial \varOmega \). Let \(m\in \mathbb {N}\) be given and \(\mathcal {R}\) a finite subset of \(\mathbb {Z}_+^m\times \mathbb {Z}_+^m\). For all \((\varvec{\alpha },\varvec{\beta })\in \mathcal {R}\) the reaction rates \(k_{(\varvec{\alpha },\varvec{\beta })}:\varOmega \times \mathbb {R}^m\rightarrow \mathbb {R}_+\) satisfy the Carathéodory condition and there exist real constants \(0<\underline{c}_{k}\), \(\overline{c}_{k}<\infty \) such that \(\underline{c}_{k} \le k_{(\varvec{\alpha },\varvec{\beta })}(\varvec{x},\varvec{y})\le \overline{c}_{k}\), f.a.a. \(\varvec{x}\in \varOmega \), \(\forall \varvec{y}\in \mathbb {R}^m\). Source terms of reactions are at most quadratic. The diffusion coefficients \(D_\nu :\varOmega \times \mathbb {R}^m\rightarrow \mathbb {R}_+\) satisfy the Carathéodory condition and there exist constants \(0<\underline{c}_{D}\), \(\overline{c}_{D}<\infty \) such that \(\underline{c}_{D}\le D_\nu (\varvec{x},\varvec{y}) \le \overline{c}_{D}\), f.a.a. \(\varvec{x}\in \varOmega \), \(\forall \varvec{y}\in \mathbb {R}^m\) and \(\nu =1,\ldots ,m\). Furthermore, \(\overline{u}_\nu \), \(U_\nu \in L^{\infty }(\varOmega )\) and there exist constants \(0<\underline{c}_{\overline{u}}\), \(\overline{c}_{\overline{u}}<\infty \) and \(0<\underline{c}_{U}\), \(\overline{c}_{U}<\infty \) such that \(\underline{c}_{\overline{u}}\le \overline{u}_\nu (\varvec{x}) \le \overline{c}_{\overline{u}}\), and \(\underline{c}_{U}\le U_\nu (\varvec{x}) \le \overline{c}_{U}\), resp. f.a.a. \(\varvec{x}\in \varOmega \) and \(\nu =1,\ldots ,m\). Finally, we assume \(\{ a \in \mathbb {R}_+^m : a^{\varvec{\alpha }}=a^{\varvec{\beta }} \,\, \forall (\varvec{\alpha },\varvec{\beta })\in \mathcal {R}, \left( \int _{\varOmega } \left( \overline{u}_\nu a_\nu - U_\nu \right) \, dx \right) _{\nu =1}^m \in \mathcal {S}\} \cap \partial \mathbb {R}_+^m = \emptyset \).
Remark 1
These technical assumptions allow us to handle a general class of reaction–diffusion systems, including heterogeneous materials and nonlinear diffusion processes. Heterogeneous materials can be found quite often in the modeling of biological or chemical processes involving different phases (see [4]), therefore we assume the dependence of the diffusion coefficients and the reaction rate coefficients on the spatial variable. The dependence of the diffusion coefficients on the state variable is motivated by problems like those considered in Sect. 4. For example, recombination reactions of Shockley–Read–Hall and Auger type, see [19, 32, 44], contain reaction rate coefficients depending on the state variable. The assumptions on the space dimension and on the reaction order are technical to obtain results in a general class of problems as done in [19] for the continuous problem. Note, that only the source terms of the reaction terms are restricted, the sink terms may be large.
2.2 Summary of known results for the continuous problem
Let \(u:=(u_1,\ldots ,u_m)\), \(v:=(v_1,\ldots ,v_m)\) and \(a:=(a_1,\ldots ,a_m)\) denote the vectors of densities, chemical potentials and activities of all species. Let us introduce the Gelfand triple \(X\Subset Y \cong Y^* \Subset X^*\), where \(X:=H^1(\varOmega ,\mathbb {R}^m)\) and \(Y:=L^2(\varOmega ,\mathbb {R}^m)\). Moreover, we define the operators \(A:X\cap L^\infty (\varOmega ,\mathbb {R}^m)\rightarrow X^*\), \(E:X\rightarrow X^*\) by
In the setting of (A1), a weak formulation of (4) can be stated as follows: Find \((u,v)\) such that:
Such problems have been investigated in various papers, see e.g. [24, 31]; the papers [19, 20, 27–29] treat also electrically charged species, such that the flux terms additionally contain drift contributions, and a Poisson equation for the self consistent calculation of the electrostatic potential is added to (4). The papers [19, 20, 25, 27] additionally deal with more general state equations than (1). We shortly summarize results for the continuous reaction–diffusion system (4) obtained in the cited papers.
By means of the stoichiometric subspace \(\mathcal {S}= {{{\mathrm{span}}}}\{\varvec{\alpha }-\varvec{\beta }:(\varvec{\alpha },\varvec{\beta })\in \mathcal {R}\}\) we define some compatibility class
For an example illustrating the meaning of \(\mathcal {R}\), \(\mathcal {S}\) and \(\mathcal {S}^{\bot }\) we refer to Sect. 4. All solutions \((u,v)\) to (P) fulfill the invariance property
Therefore, if \(u^*:=\lim \limits _{t\rightarrow \infty }u(t)\) exists, then we have necessarily \(u^*\in U+\mathcal {U}.\) According to [19, 27, 28] there exists a unique stationary solution \((u^*,v^*)\) to (P) additionally fulfilling \(u^*\in U+\mathcal {U}\). This \((u^*,v^*)\) is a thermodynamic equilibrium of the system. Along solutions to the nonstationary problem (P) the free energy
decays monotonously and exponentially to its equilibrium value \(F(u^*)\),
with \(\lambda >0\) depending only on the data, see [25, 27–29].
If all reactions exhibit source terms of maximal order 2 then all solutions \((u,v)\) to (P) are globally bounded, especially the particle densities are positively bounded away from zero (see e.g. [29]).
Nevertheless, in special cases, using the concrete structure of the underlying reaction system also some systems not fulfilling the general formulated condition of source terms of maximal order 2 can be handled e.g. under the ’intermediate sum condition’ (see [37]), where a priori estimates for positive linear combinations of densities are obtained or in the case of cluster reactions of higher order (see [30]) where in the proof for the a priori estimates simultaneously different powers of the chemical activities of the different species are used as test functions.
Introducing suitable regularized problems, finding a priori estimates which do not depend on the regularization level, and solving the regularized problems, the existence of solutions to (P) is shown in [19]. Uniqueness results for (P) can be obtained by classical arguments (see [29]), if the diffusion coefficients do not depend on the state variables. For cases with diffusion coefficients depending on the state variable we refer to [19].
Let us remark that in three space dimensions there are similar results available, but stronger restrictions on the reactions are needed: reactions of maximal order three can be handled to obtain the exponential decay of the free energy (see [31]). Global upper and lower bounds of the solution and solvability of problem (P) can be proven under the restriction that the order of the source terms in each equation have to be less or equal to \(\frac{5}{3}\) (see [19]).
3 Discretized reaction–diffusion systems
3.1 Voronoi finite volume discretization
In the previous part we saw that the solution of reaction–diffusion systems preserve some quantities like invariants (7) and positivity. Therefore, the aim is to respect the conservation of these properties by the approximated solution. Conservation of qualitative properties like local and global mass conservation, maximum principles, positivity or more generally, \(L^{\infty }\) bounds is known to be difficult for finite element discretizations, see e.g. [5, 36]. One important reason for that is that finite element methods do usually not allow to test the discrete equations with the positive part of a finite element function, since it does not lie in the discrete test space space, in general. On the contrary, finite volume methods allow such test functions [14, 21], and we will show how the use of appropriate discrete test functions will deliver us the desired conservation of qualitative properties. The finite volume method has been developed by engineers to study systems of conservation laws.
In the following, we work with Voronoi meshes, which represent one class of admissible finite volume meshes [12]. Our notation is basically taken from [24] and visualized in Fig. 1.
Let \(\varOmega \) be an open bounded, polyhedral subset of \(\mathbb {R}^2\). A Voronoi mesh is defined as triple \(\mathcal {M}=(\mathcal {P},\mathcal {V},\mathcal {E})\). Here, \(\mathcal {P}\) denotes a family of grid points in \(\bar{\varOmega }\), \(\mathcal {V}\) denotes a family of Voronoi control volumes and \(\mathcal {E}\) denotes a family of faces in \(\mathbb {R}\). The number of grid points is denoted by \(M=\#\mathcal {P}\).
The corresponding control volume \(K\) of each grid point \(x_K\in \mathcal {P}\) is defined by
The set of all neighboring control volumes of \(K\) is denoted by \(\mathcal {N}_{\mathcal {V}}(K)\). The Lebesgue measure of each control volume \(K\) is denoted by \(\mathord {\left| K\right|}\) and the mesh size of \(\mathcal {M}\) by
For two different \(K,\,L\in \mathcal {V}\) the one-dimensional Lebesgue measure of \(\overline{K}\cap \overline{L}\) is either zero or \(\overline{K}\cap \overline{L}=\overline{\sigma }\) for one \(\sigma \in \mathcal {E}\). Here the symbol \(\sigma =K|L\) denotes the one-dimensional face between the control volumes \(K\) and \(L\) and \(m_\sigma \) is its Lebesgue measure.
We introduce the subset \(\mathcal {E}_{int}\subset \mathcal {E}\) containing all interior faces. Further, we introduce for all \(K\in \mathcal {V}\) the subset \(\mathcal {E}_K \subset \mathcal {E}_{int}\), such that \(\forall \sigma \in \mathcal {E}_K\, \exists \, L\in \mathcal {N}_{\mathcal {V}}(K): \sigma = \overline{L} \cap \overline{K}\).
The Euclidian distance between two neighboring grid points \(x_K\), \(x_L\in \mathcal {P}\) over the face \(\sigma =K|L\in \mathcal {E}_{int}\) is denoted by \(d_\sigma \).
Definition 2
(see [24]) Let \(\varOmega \) be an open bounded, polygonal subset of \(\mathbb {R}^2\) and \(\mathcal {M}=(\mathcal {P},\mathcal {V},\mathcal {E})\) a Voronoi mesh.
-
The symbol \(X_{\mathcal {V}}(\mathcal {M})\) denotes the set of all piecewise constant functions from \(\varOmega \) to \(\mathbb {R}\) which are constant on every Voronoi control volume \(K\in \mathcal {V}\). The constant value of \(w_h\in X_{\mathcal {V}}(\mathcal {M})\) on the control volume \(K\in \mathcal {V}\) is denoted by \(w_K\).
-
Let \(p\ge 1\). The discrete \(L^p-\) norm of \(w_h\in X_{\mathcal {V}}(\mathcal {M})\) is defined by
$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}= \left( \sum _{K\in \mathcal {V}}\mathord {\left| K\right|} \mathord {\left|w_K\right|}^p\right) ^{1/p}. \end{aligned}$$ -
The discrete \(H^1\) semi-norm of \(w_h\in X_{\mathcal {V}}(\mathcal {M})\) is defined by
$$\begin{aligned} \mathord {\left|w_h\right|}_{H^1,\mathcal {M}}^2 = \sum _{\sigma =K|L\in \mathcal {E}_{int}} T_{\sigma } \mathord {\left|w_K-w_L\right|}^2,\quad T_{\sigma }:=\frac{m_\sigma }{d_\sigma }. \end{aligned}$$Here \(w_K\) and \(w_L\) are the constant values of \(w_h\) in the control volumes \(K\) and \(L\). The term \(T_\sigma \) is the so called transmissibility across the edge \(K|L\). The usual \(H^1-\) norm is given by \(\mathord {\left||w_h\right||}_{H^1,\mathcal {M}}^2=\mathord {\left|w_h\right|}_{H^1,\mathcal {M}}^2+ \mathord {\left||w_h\right||}_{L^2}^2\).
We prescribe the approximation of a function \(f:\varOmega \times \mathbb {R}^m\rightarrow \mathbb {R}\) by
where \(K\in \mathcal {V}\). In this context we introduce the approximation of the diffusion coefficients and the reaction terms on a Voronoi cell \(K\in \mathcal {V}\) by
and analogously the approximation of \(k_{(\varvec{\alpha },\varvec{\beta }) K}\). The corresponding piecewise constant function can be estimated from above and below by the upper and lower bound of the continuous function. We define the reference density \(\overline{u}_{\nu K}\) and the density \(u_{\nu K}\) being constant on a control volume \(K\in \mathcal {V}\), and the mass \(u_{\nu }^{(K)}\) of the \(\nu -\)th species in \(K\in \mathcal {V}\) by
For every species \(\mathsf X_{\nu }\), \(\nu =1,\ldots ,m\), we introduce the discrete initial values by
The space-discrete version of the continuous problem (P) is obtained by testing with the characteristic function of \(K\). Using Gauss theorem, we derive the approximated flux term
where
represents the logarithmic mean value of \({{\mathrm{e}}}^x\) in the interval \([x,y]\). In the following we write \(Z_{\nu }^{\sigma }=Z(v_{\nu L},v_{\nu K})\) for \(\sigma =K|L\in \mathcal {E}_{int}\) and \(D_{\nu K}=D_{\nu K}({{\mathrm{e}}}^{v_{1 K}},\ldots , {{\mathrm{e}}}^{v_{m K}})\). With this definition of \(Z_{\nu }^{\sigma }\) it is possible to switch between a gradient in potentials and activities, i.e. the discrete version of \(\nabla a_{\nu }=a_{\nu } \nabla v_{\nu }\) holds. The symbol \(Y_{\nu }^{\sigma }\) defines some averaging of \(D_\nu \overline{u}_\nu \) over the edge \(\sigma =K|L\), which is symmetric in \(K\) and \(L\). Possible averagings are, e.g.,
For another averaging which is exact along an aligned edge we refer to [11]. In the sequel all results are independent of the particular chosen \(Y_{\nu }^{\sigma }\).
Following [23] we use the notation
Furthermore, we define the scalar products
Definition 3
(Time discretization) A time discretization of \(\mathbb {R}_+\) is defined as a strictly increasing sequence of real numbers \((t_n)_{n\in \mathbb {N}}\subset \mathbb {R}_+\) with \(t_0=0\) and \(t_n\rightarrow \infty \) for \(n\rightarrow \infty \). The time step is defined by
A discretization of the whole domain \(Q=\varOmega \times \mathbb {R}_+\) is defined by the tuple \(\mathcal {D}=(\mathcal {M},(t_n)_{n\in \mathbb {N}})\). To indicate the relation to the continuous problem (P) we give a variational form of the discrete problem. We introduce the operator \(\widehat{E}:\mathbb {R}^{Mm}\rightarrow \mathbb {R}_{+}^{Mm}\) by
which maps in every control volume the chemical potential of the species to its mass. Furthermore we define \(\widehat{A}:\mathbb {R}^{Mm}\rightarrow \mathbb {R}^{Mm}\) by
Using these definitions we can state the discrete version of (P) by: Find a tuple \((\varvec{u},\varvec{v})\) such that
As long as the meaning is clear, we will neglect in the following the current time step \(t_n\) and we shortly write \(u_{\nu K}\) instead of \(u_{\nu K}(t_n)\). Using this notation, for every time step, in every control volume \(K\in \mathcal {V}\) and \(\nu =1,\ldots ,m\) we have to solve the following fully implicit system of nonlinear equations
with \(u_{\nu K}(t_n)=\overline{u}_{\nu K}{{\mathrm{e}}}^{v_{\nu K}}\). The discrete variational form of \(\widehat{A}\) is given by
We associate the discrete (vectorial) solutions \((\varvec{u,\varvec{v}})\) to \((P_{D})\) to piecewise constant functions \((u_h,v_h)\) and call them solutions to \((P_{D})\), too.
Remark 2
In all proofs, real constants \(C>0\) with different meaning are numbered consecutively. The constants only depend on the data (lower bounds of the diffusion coefficients, upper bounds of the reaction rate constants, lower and upper bounds of the reference densities and initial values, see (A1) and not on the discretization, unless otherwise stated.
Concerning vectors \(\varvec{w}\in \mathbb {R}^k\), \(k\in \mathbb {N}\), we use: Writing \(\varvec{w}\ge \varvec{0}\) we mean \(w_i\ge 0\) resp. \(w_i> 0\) for \(i=1,\ldots ,k\). By \(\ln \varvec{w}\) we denote \((\ln w_i)_{i=1}^{k}\), and by \({{\mathrm{e}}}^{\varvec{w}}\) the vector \(({{\mathrm{e}}}^{w_i})_{i=1}^{k}\).
Moreover the symbols \(S_1\), \(S_2\) and \(S_3\) have a local meaning and differ from time to time. Generally these terms arise from testing the problem \((P_{D})\) by test functions and discussing the expressions for the time-derivative (\(S_1\)), diffusion term (\(S_2\)) and reaction term \((S_3)\) separately.
Remark 3
For obtaining upper and lower bounds of the concentration of the problem \((P_{D})\) we use a technique introduced by Moser [38] which needs an appropriate, initial a-priori bound in some \(L^p\), \(p\ge 1\) space. These starting points of the bootstrapping procedures are provided by physically motivated energy estimates (see Lemmas 2 and 3) proven in [24] in the setting of Voronoi finite volume discretizations. If the results were provided on admissible finite volume meshes [12, Definition 3.1], it would be possible to prove similar bounds of the concentrations, too.
3.2 Local existence result
In analogy to the continuous setting we define the subspaces
and
Another characterization of \(\widehat{\mathcal {U}}^{\bot }\) is given by
Using (11), one easily finds \(\mathord {\left\langle \widehat{A}\varvec{v},\varvec{v}^{\bot }\right\rangle }=0\quad \forall \varvec{v}^{\bot }\in \widehat{\mathcal {U}}^{\bot }\text { and } \forall \varvec{v}\in \mathbb {R}^{Mm}\), hence the solutions \((\varvec{u},\varvec{v})\) to \((P_{D})\) fulfill the invariance property
Next we define the operator \(\widehat{B}:\mathbb {R}^{Mm}\rightarrow \mathbb {R}^{Mm}\), by
for all \(\varvec{u}\in \mathbb {R}^{Mm}\). Have in mind that \(u_{\nu K}= u_{\nu }^{(K)}/\mathord {\left| K\right|}\) holds. The solvability of \((P_{D})\) can be proved by the investigation of the solvability of the following problem: Find a positive \(\varvec{u}\in \mathbb {R}^{Mm}\) such that
The relation between \(\widehat{A}\) and \(\widehat{B}\) is given by
In particular, the invariance property (14) also holds for solutions to \((P_{\tilde{D}})\).
Next, we prove existence under additional assumptions with respect to the reaction terms. We assume that
and
Condition (15) is known as quasi positivity, see [2, 41]. The second condition (16) imposes conservation of atom number, see [22], \((\)Th\(_2)\)]. From the quasi positivity we deduce for one \(u_{\nu K}=0\), \(K\in \mathcal {V}\) and \(\nu =1,\ldots ,m\) that
which means that zero concentrations are raised by the system.
Lemma 1
Let (A1), (15), and (16) be fulfilled. Moreover let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a discretization and let \(\varvec{\tilde{u}} \in \widehat{\mathcal {U}} + \varvec{U}\) with \(\varvec{\tilde{u}} > \varvec{0}\) be fulfilled. Then for all \(s > 0\), there exists an \(\varvec{u} \in \mathbb {R}^{Mm}\) such that
and \(\varvec{u} >\varvec{0}\). Furthermore \(\varvec{u}\in \widehat{\mathcal {U}}+\varvec{U}\).
Proof
In the following, we use Brouwer’s fixed point theorem to deduce the existence of a solution. We define the set of all densities which fulfill the same invariants as the initial concentration \(\varvec{U}\)
The main point of the proof is to show that the fixed point is positive. Because \(\mathcal {C}\) is the intersection of an affine space with \(Mm\) half spaces of nonnegative densities, the space \(\mathcal {C}\) is convex. By construction, \(\mathcal {C}\) is also a closed set. Since all elements are nonnegative and fulfill (14), the set \(\mathcal {C}\) is bounded. Also by construction, the boundary of \(\mathcal {C}\) possesses at least one component that is zero. From (16), we deduce the existence of a vector \(\varvec{s}^{\bot } \in \mathcal {S}^{\bot }\), with only positive entries. As a consequence of (13), we conclude the existence of \(\varvec{w}^{\bot } \in \widehat{\mathcal {U}}^{\bot }\), with only positive entries. Since \(\varvec{\tilde{u}} \in \mathcal {C}\) holds, we find by (14) and (13) that
We define \(\theta : \mathcal {C}\times [0, \infty ) \rightarrow \mathbb {R}\) by
Since \(\varvec{\tilde{u}} \in \mathcal {C}\) holds, and \(\mathcal {C}\) is bounded, the function \(\theta \) is well defined. Using the convexity of \(\mathcal {C}\), we deduce the continuity of \(\theta \). Hence, the function \(\varphi _s: \mathcal {C}\rightarrow \mathcal {C}\) with
is continuous for every \(s > 0\), and the function \(\theta (\varvec{u}, s)\) ensures that \(\varphi _s(\varvec{u}) \in \mathcal {C}\) holds.
Using Brouwer’s fixed point theorem, we conclude the existence of a nonnegative fixed point \(\varvec{u}\) of \(\varphi _s\) for all \(s> 0\). Assuming one or more components of \(\varvec{u}\) are zero, then by (15) and (17) we find that these components of \(-\theta (\varvec{u}, s) \widehat{B} \varvec{u}\) are nonnegative, which leads to a contradiction to \(\varvec{\widetilde{u}}>\varvec{0}\). Therefore, the fixed point is not only nonnegative, but positive. But then, the fixed point is not on the boundary of \(\mathcal {C}\) and \(\theta (\varvec{u}, s) = s\) must hold. This means that the fixed point of (19) is also a fixed point of (18). \(\square \)
By induction we conclude:
Theorem 1
Let (A1), (15), and (16) be fulfilled. Moreover let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a discretization and let \(\varvec{U}=\varvec{u}(t_0) > \varvec{0}\) be fulfilled. For all \(t_n>0\) there exists at least one solution \(\varvec{u}(t_n) > \varvec{0}\) with \(\varvec{u}(t_n)\in \widehat{\mathcal {U}}+\varvec{U}\) of the nonlinear equation \((P_{\tilde{D}})\).
This implies:
Theorem 2
Let (A1), (15), and (16) be fulfilled. Moreover let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a discretization and let \(\varvec{U}=\varvec{u}(t_0) > \varvec{0}\) be fulfilled. Then there exists a solution \((\varvec{u}(t_n),\varvec{v}(t_n))\) of the discrete Problem \((P_{D})\). Moreover there exists a unique stationary solution \((\varvec{u}^*,\varvec{v}^*)\) of \((P_{D})\) with \(\varvec{u}^{*}\in \widehat{\mathcal {U}}+\varvec{U}\) and \(0<c\le \varvec{u}^{*}\).
Proof
Since \(\varvec{u}(t_n)>0\) and \(\varvec{v}(t_n)=\ln (\varvec{u}(t_n)/\varvec{\overline{u}})\), the solution of (18) delivers a solution to \((P_{D})\), too. From [23, Theorem \(3.1\)] we conclude the existence and uniqueness of the stationary solution. We recall that in our case the diffusion coefficients may depend on the state in contrast to [23]. But a careful inspection of the proof given there, shows the validity of the result for this situation, too. \(\square \)
Remark 4
Local existence results for systems with reaction terms not fulfilling (15) and (16) can be proven by investigating a “regularized” problem which arises from \((P_{D})\) by cutting off the nonlinearities in a suitable way at a certain level and using the theory of pseudomonotone operators, see e.g. [19, 29, 45]. Similarly, in the proof of Lemma 1 we cut off the time step \(s\) by using the function \(\theta (\cdot ,\cdot )\).
3.3 Physically motivated estimates
In this section we show that physical motivated arguments lead to a priori estimates for the solutions to \((P_{D})\). These estimates deliver a starting point for the Moser iteration. We introduce the free energy being a convex functional. Since we consider an isolated process we expect the decay along trajectories. In the literature the term free energy is often denoted as entropy [6, 7]. All results are based on the articles [24, 25]. We also refer to [9] for basic notation and results from convex analysis.
First we introduce the discrete potential \(\widehat{\varPhi }:\mathbb {R}^{Mm}\rightarrow \mathbb {R}\) as the potential of \(\widehat{E}\) by
Due to \(\varvec{u}=\widehat{E}\varvec{v}\), it holds \(\varvec{u}=\widehat{\varPhi }^{\prime }(\varvec{v})\). The conjugate functional of \(\widehat{\varPhi }\) is defined by \(\widehat{F}:\mathbb {R}^{Mm}\rightarrow \overline{\mathbb {R}}\),
For a given argument \(\varvec{u}\in \mathbb {R}^{Mm}\) the value of \(\widehat{F}(\varvec{u})\) can be interpreted as the free energy of the state \(\varvec{u}\). Together with \(\varvec{u}=\widehat{E}\varvec{v}\) we find
Using the elementary inequality \(\ln s\ge 1-1/s\), \(s>0\) we observe the nonnegativity of the free energy
Finally we introduce the discrete dissipation functional \(\widehat{D}:\mathbb {R}^{Mm}\rightarrow \mathbb {R}\) by
As a consequence of (11) we mention that for all \(v\in \mathbb {R}^{Mm}\)
Lemma 2
(Monotonous decay of the free energy) Let (A1) be fulfilled and let \((u_h,v_h)\) be a solution to \((P_{D})\) for a discretization \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\). Then for \(0\le t_{n_1}<t_{n_2}\in \mathbb {R}_{+}\) holds
i.e., the free energy decays along all solutions to \((P_{D})\). Moreover, it holds
Proof
The monotonicity of the free energy is proven in [24, Lemma 3.1]. Using the elementary inequalities
we get
\(\square \)
Remark 5
Conservation of atoms (16) implies an alternative possibility to obtain a priori bounds in \(L^1\) without using the free energy of the system. From (16), it results the existence of a vector \(\varvec{s}^{\bot }\in \mathcal {S}^{\bot }\) with \(s^{\bot }_{\nu }>0\), \(\nu =1,\ldots ,m\). From (13) we deduce a \(\varvec{w}^{\bot } \in \widehat{\mathcal {U}}^{\bot }\) with \(w_{\nu K}^{\bot }=s^{\bot }_{\nu }>0\) for all \(K\in \mathcal {V}\) and \(\nu =1,\ldots ,m\). By (14) and \((P_{D})\) we obtain for all \(N\ge 0\)
We refer to [41] for more examples of systems fulfilling this property and for generalizations.
3.4 Global upper bounds
In this section we want to prove global upper bounds for the densities that are uniform for all discretizations fulfilling the additional assumption:
-
(A2)
There exists a constant \(\overline{t}_{\delta }<\infty \) such that the largest possible time step is bounded by \(\max \limits _{n\in \mathbb {N}} t_\delta ^{(n)}\le \overline{t}_\delta \) for all considered time discretizations.
For obtaining the global bounds, we use a bootstrapping technique introduced by Moser, see [38], and apply it to the discretized problem.
Theorem 3
( Upper bounds) Let (A1) be fulfilled and let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a class of discretizations fulfilling (A2). Then there exists a constant \(c_1>0\) only depending on the data and not on \(\mathcal {D}\) such that for every solution \((u_h,v_h)\) to \((P_{D})\)
holds uniformly for all discretizations \(\mathcal {D}\). Furthermore there exists a second constant \(c_2>0\) only depending on the data and not on \(\mathcal {D}\) such that
holds uniformly for all discretizations \(\mathcal {D}\).
Remark 6
In the continuous case one uses test functions
for \(p=2^k\), \(k\ge 1\) for problem (P) to obtain global upper bounds of the solutions, see [29, Lemma \(4.1\) and Theorem \(4.2\)]. The discrete proof follows the continuous counterpart. However, since we have a time discretization, the test functions have to be modified and we have to estimate some error terms coming from the discretization of the diffusion term.
Proof
First we mention that
with \(\overline{t}_\delta \) given in (A2). We introduce \(z_{\nu ,h}=({{\mathrm{e}}}^{v_{\nu ,h}}-\kappa )^{+}\) with
and \(w_{\nu ,h}=z_{\nu ,h}^{p/2}\), \(p\ge 2\). The constant \(\kappa \) is chosen in such a way that \(z_{\nu ,h}(t_0)=(U_{\nu ,h}/\overline{u}_{\nu ,h}-\kappa )^+=0\). Now, we test \((P_{D})\) with test functions \(p e^{t_{n-1}} z_{\nu ,h}^{p-1}(t_n)\), \(p\ge 2\), and obtain
with
Next, the expressions for the time-derivative (\(S_1\)), diffusion term (\(S_2\)) and reaction term \((S_3)\) are estimated separately.
Time derivative: Straightforward calculations, using that the product of the positive and negative part of a function vanishes, yield
Using (42) and the fact that \(z_{\nu K}^{p-1}(t_{n})({{\mathrm{e}}}^{v_{\nu K}(t_{n-1})}-\kappa )^{-}\ge 0\) holds, we get
In the last line we used \(z_{\nu ,h}(t_0)=0\) and (20).
Diffusion term: Now, we consider the diffusion term \(S_2\). Applying the definition (9) to
using the lower bound of the reference densities and the diffusion coefficients, inequality (41), the notation \(w_{\nu ,h}=z_{\nu ,h}^{p/2}\), and the extension of the \(H^1\) semi-norm to the full \(H^1\) norm, we find for \(p\ge 2\)
In the last two lines we set \(\underline{c}_{D \overline{u}}=\underline{c}_{D}\underline{c}_{\overline{u}}\) and use \(2\le 4(p-1)/p\le p\) for \(p\in [2,\infty )\).
Reaction terms: Together with (5), the inequalities
Muirhead’s inequality (see [39])
and \(x^{p}\le x^{p+1}+1\) for \(x\ge 0\) and \(p\ge 1\) we can estimate the reaction terms (3) with at most quadratic source terms (see (A1) and (5)) by
with constants \(C_1\), \(C_2>0\).
The tested equation: Using the obtained estimates of the three terms together with \(S_1=S_2+S_3\), leads to
with a constant \(C_3>0\) which can be chosen such that it depends on the largest possible time step \((\)see (A2) and (21)\()\) and the data, but not on \(p\).
Bounds in \(L^2\): For obtaining the \(L^2\) bound, we set \(p=2\). The last term in (22) (the \(L^{3}\) norm of \(z_{\nu ,h}\)) can be controlled by the discrete Gagliardo–Nirenberg inequality (39), i.e. we find for all \(\epsilon >0\) a \(c_{\epsilon ,3}>0\) such that
We continue (22) by
where \(g_1(\epsilon ):=-2\underline{c}_{D \overline{u}}+4\epsilon C_3\mathord {\left||z_{\nu ,h}\ln z_{\nu ,h}\right||}_{L^1}\). To choose the constant \(\epsilon \) we have to control the \(L^1\) norm of \(z_{\nu ,h}\ln z_{\nu ,h}\) and \(z_{\nu ,h}\). From Lemma 2 and \(z_{\nu ,h}\le a_{\nu ,h}\) we deduce the boundedness of \(\mathord {\left||z_{\nu ,h}\right||}_{L^1}\). Since \(\mathord {\left|(x-\mu )\ln (x-\mu )\right|}\le x \ln x +1\) holds for \(x\ge \mu \ge 0\) and by Lemma 2 we obtain
We fix the constant \(\epsilon >0\), coming from the Gagliardo–Nirenberg inequality such that \(g_1(\epsilon )\le 0\) holds. From (20) we get
and from (22) for \(p=2\) we can derive the boundedness of
with a constant \(C_4>0\). The first result of the theorem follows by using the inequality \(u_{\nu K}/\overline{u}_{\nu K}\le z_{\nu ,K}+\kappa \).
Moser iteration for \(p\ge 4\): For \(p\ge 4\) let \(r=\frac{2(p+1)}{p}\) be introduced. Note that \(r\in (2,5/2]\) for \(p\in [4,\infty )\). Using \(w_{\nu ,h}=z_{\nu ,h}^{p/2}\), the estimate (22) can be written as
By the discrete Gagliardo–Nirenberg inequality (35) we obtain
For \(r\in (2,\,5/2]\) the constants appearing in the Gagliardo–Nirenberg inequality (35) can be uniformly bounded by
Using Young’s inequality with \(q=\frac{2p}{p+2}\), \(q^\prime =\frac{2p}{p-2}\) and \(\epsilon >0\) we get
Inserting this in (24) we find
with \(g_2(\epsilon )=-2\underline{c}_{D \overline{u}}+ C_3 \overline{c}_r \epsilon \frac{p}{q}\). The constant \(\epsilon \) is fixed such that \(g_2(\epsilon )=0\), i.e.
Then the term in front of the \(L^{1}\) norm can be decomposed into two factors
The first factor is bounded and for the second factor we find for \(p\ge 4\) by monotonicity
Therefore we define
and we proceed with
Therefore, with some constant \(C_5>1\) we get
Iteratively using this inequality and setting \(p=2^k\), \(k\ge 2\) and
we find for \(k\in \mathbb {N}_+\), \(k\ge 2\) the recursion formula \(b_k \le (4)^{k} C_5 (b_{k-1})^{2\frac{2^{k-1}}{2^{k-1}-1}}\) and conclude
By induction one can prove
see [29, p. \(217\)]. The product \(\theta =\prod _{j=1}^{\infty } \ \frac{2^{j}}{2^{j}-1}\) is finite and \(b_k \le (16 C_5 b_1)^{\theta 2^k}\). Since \(b_1\) is bounded from above by (23) we obtain for \(k\ge 2\)
and finally with [33, Theorem \(2.11.5\)]
for \(k\rightarrow \infty \). From \(u_{\nu ,h}/\overline{u}_{\nu ,h} \le z_{\nu ,h}+\kappa \) the result follows. \(\square \)
3.5 Asymptotics
In this section we will extend the result of Lemma 2. We mention the result of [24] where it is proved that the free energy decays exponentially along trajectories. We also note that in special situations an explicit rate of convergence is proven, see [6].
Lemma 3
(Exponential decay, see [24, Lemma 3.1 & Theorem 3.3]) Let (A1) be fulfilled and let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a class of discretizations fulfilling (A2). Then there exist constants \(\lambda >0\) and \(c>0\) only depending on the data and not on \(\mathcal {D}\) such that for every solution \((u_h,v_h)\) to \((P_{D})\)
uniformly for all discretizations \(\mathcal {D}\).
Proof
The complete proof is given in [15, Theorem 3.9.3]. Instead of using the discrete Sobolev–Poincaré inequality [26, Theorem 2.2] we use (37) in the estimate of [24], \((3.12)\)], in order to avoid an additional assumption on the mesh needed in [26, Theorem 2.2]. Additionally, the proof of [15, Theorem 3.9.3] does not need any restriction on the mesh size, which would result from the proof of [24, Theorem 3.5]. \(\square \)
Using the \(L^\infty \) bounds from Theorem 3 we can prove the following result.
Corollary 1
(Asymptotics of the solution) Let (A1) be fulfilled and let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a class of discretizations fulfilling (A2), moreover let \((u_{h}^{*},v_{h}^{*})\) be the thermodynamic equilibrium to \((P_{D})\), see Theorem 2. Then there exist constants \(\lambda _p>0\) and \(c>0\) only depending on the data and not on \(\mathcal {D}\) such that for every solution \((u_h,v_h)\) to \((P_{D})\) and \(p\in [1,\infty )\) the estimate
holds uniformly for all discretizations \(\mathcal {D}\).
Proof
Using Hölders inequality we find
First, we note that
As a consequence of Theorem 3, we obtain the boundedness of
and find by Lemma 3 the desired estimate (27). \(\square \)
3.6 Global lower bounds
Now, we intend to show uniform global lower bounds of the densities or in other words upper bounds of the negative part of the chemical potentials. In the continuous setting, this was done in [19] and [29, p. \(18\)]. In a first step we need lower bounds in \(L^1\) which provide a suitable start for the Moser iteration.
Lemma 4
(Lower bounds in \(L^1\)) Let (A1) be fulfilled and let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a class of discretizations fulfilling (A2). Then there exists a constant \(c_1>0\) only depending on the data and not on \(\mathcal {D}\) such that for every solution \((u_h,v_h)\) to \((P_{D})\)
holds uniformly for all discretizations \(\mathcal {D}\).
Proof
Following [29, p. \(18\)] we define the convex and lower semicontinuous functional \(\widehat{\varTheta }:\mathbb {R}^{m}\rightarrow \bar{\mathbb {R}}\) by
and its conjugate convex functional
which may be written in the explicit form
Here \(\varvec{u}^{*}\) is the unique stationary solution to \((P_{D})\) according to Theorem 2. We introduce \(\overline{\varvec{z}}_{\nu }:=((1-u^{*}_{\nu K}/u_{\nu K})^{-})_{K\in \mathcal {V}}\) and the corresponding \(\overline{z}_{\nu ,h}\in X_{\mathcal {V}}(\mathcal {M})\), and observe that \(-\overline{\varvec{z}}_{\nu }\in \partial \widehat{G}(\varvec{u}_\nu )\). Testing the discrete problem \((P_{D})\) with the test function \((0,\ldots ,0,-\overline{\varvec{z}}_\nu ,0,\ldots ,0)\) leads to \(S_1=\sum _{n=1}^{N} t_{\delta }^{(n)}(S_2+S_3)\) with
Using the subdifferential property of \(\widehat{G}\) we find
Both terms \(S_2\) and \(S_3\) will be estimated on the subsets
Diffusion: In a first step we show that the diffusion term \(S_2\) is negative. We remark that \(\overline{z}_{\nu K}=0\) for all \(K\in \varOmega _{+}(t_n)\). Using (9) and \(a_{\nu K}^{*}=a_{\nu L}^{*}\) \(\forall K,\,L\in \mathcal {V}\), we find \(S_2=S_{21}-S_{22}\) with
Using \((x-y)(1/x-1/y)\le 0\) for all \(x,y>0\) we get \(S_{21}\le 0\). If \(K\in \varOmega _{-}(t_n)\) holds, the term \(a_{\nu K}^{*}-a_{\nu K}\) is positive. On the other hand if \(L\in \varOmega _{+}(t_n)\) we have \(a_{\nu L}-a_{\nu L}^{*}> 0\) and therefore \(S_{22}\ge 0\) and finally \(S_2\le 0\).
Reactions: On \(\varOmega _{+}\) reaction terms multiplied by the test function vanish. Since \((\varvec{\alpha }-\varvec{\beta })\cdot \varvec{v}_{K}^{*}=0\) \(\forall K\in \mathcal {V}\), we find on \(\varOmega _{-}(t_n)\)
with
The terms in the parentheses of \(S_{31 K}\) are Lipschitz continuous in \((\varvec{u}_{K}/\varvec{u}_{K}^*)\) on \([0,R]^{m}\), \(R>0\) and have at \((1)_{\nu =1}^{m}\) the value \(0\). Together with the global (upper) boundedness of \((\varvec{u}_{K}/\varvec{u}_{K}^*)\) we obtain
We consider two cases: Note that \(1<u_{\nu K}^{*}/u_{\nu K}\) for all \(K\in \varOmega _{-}(t_n)\). For \(\alpha _{\nu }>\beta _{\nu }\) the summands in \(S_{32 K}\) can be estimated by
This term is also Lipschitz continuous in \((\varvec{u}_{K}/\varvec{u}_{K}^*)\) on \([0,R]^{m}\), \(R>0\), and has at \((1)_{\nu =1}^{m}\) the value \(0\). Together with the global boundedness of \((\varvec{u}_{K}/\varvec{u}_{K}^*)\) we obtain the bound
The case \(\alpha _{\nu }<\beta _{\nu }\) can be handled analogously. Therefore there exists a constant \(C_4>0\) such that
\(L^1\)-estimate: From (31) together with \(S_1=\sum _{n=1}^{N} t_{\delta }^{(n)}(S_2+S_3)\) and the estimates of \(S_2\) and \(S_3\) we conclude
From Corollary 1 and Lemma 3 we conclude that
with the constant \(\lambda >0\) given in Lemma 3. Hence there exists a constant \(C_6>0\) such that \(\widehat{G}(\varvec{u}_{\nu }(t_N))\le C_6\). Let \(\underline{c}_{u^{*}}=\min _{\nu =1,\ldots ,m}{{\mathrm{ess}}}\inf _{x\in \varOmega } u_{\nu }^{*}(x)\). Together with
and the explicit form of \(\widehat{G}\) in (30) we find for all \(N\ge 0\)
from which the bounds in \(L^1\) follow. \(\square \)
Now, we show uniform global lower bounds for the chemical potentials by Moser iteration.
Theorem 4
(Lower bounds in \(L^\infty \)) Let (A1) be fulfilled and let \(\mathcal {D}=(\mathcal {M},(t_{\delta }^{n})_{n\in \mathbb {N}})\) be a class of discretizations fulfilling (A2). Then there exists a constant \(c>0\) only depending on the data and not on \(\mathcal {D}\) such that for every solution \((u_h,v_h)\) to \((P_{D})\)
holds uniformly for all discretizations \(\mathcal {D}\).
Remark 7
The proof is based on Moser iteration, too. In [29, Lemma \(4.2\) and Theorem \(4.3\)] this technique was applied to the continuous case. For \(p=2^k\), \(k\ge 1\) one takes the test function which has the \(\nu -\)th component \(-p{{\mathrm{e}}}^t z_{\nu }^{p-1} {{\mathrm{e}}}^{-v_{\nu }(t)}\) with \(z_{\nu }=(v_{\nu }+\kappa )^{-}\), \(\kappa =\max _{\nu =1,\ldots ,m}\mathord {\left||(v_{\nu }(0))^{-}\right||}_{L^\infty }\), the other components are zero. As already mentioned, in the discrete case the test functions have to be modified.
Proof
Let \(z_{\nu ,h}=(v_{\nu ,h}+\kappa )^{-}\) and \(w_{\nu ,h}=z_{\nu ,h}^{p/2}\). The constant \(\kappa \) is defined by
and therefore \(z_{\nu ,h}(t_0)=0\) holds. For \(p\ge 2\) we test the discrete Problem \((P_{D})\) with test functions which have the \(\nu \)-th component
the other components are zero. Doing this we derive \(S_1=S_2+S_3\) with
We estimate the different parts separately:
Time derivative: Since \({{\mathrm{e}}}^{x}-{{\mathrm{e}}}^{y}={{\mathrm{e}}}^{\xi }(x-y)\) holds for some \(\xi \in [x,y]\subset \mathbb {R}\) and \(z_{\nu K}^{p-1}(t_n)(v_{\nu K}(t_{n})+\kappa )^{+}=0\) we find
In the following we consider the two cases:
-
1.
From \(u_{\nu K}(t_n)> u_{\nu K}(t_{n-1})\) we get \(z_{\nu K}(t_n) \le z_{\nu K}(t_{n-1})\) and \({{\mathrm{e}}}^{\xi _{\nu K}} < {{\mathrm{e}}}^{v_{\nu K}(t_{n})}\), hence
$$\begin{aligned} I_{\nu K}(t_n)\ge p {{\mathrm{e}}}^{t_{n-1}} z_{\nu K}^{p-1}(t_n) \overline{u}_{\nu K} \left( z_{\nu K}(t_n)-z_{\nu K}(t_{n-1})\right) . \end{aligned}$$ -
2.
From \(u_{\nu K}(t_n)< u_{\nu K}(t_{n-1})\) we find \(z_{\nu K}(t_n) \ge z_{\nu K}(t_{n-1})\) and \({{\mathrm{e}}}^{\xi _{\nu K}} > {{\mathrm{e}}}^{v_{\nu K}(t_{n})}\), hence
$$\begin{aligned} I_{\nu K}(t_n)\ge p {{\mathrm{e}}}^{t_{n-1}} z_{\nu K}^{p-1}(t_n) \overline{u}_{\nu K} \left( z_{\nu K}(t_n)-z_{\nu K}(t_{n-1})\right) . \end{aligned}$$
Together with (42) and (20) we estimate \(S_1\) by
where the second line is a telescoping sum.
Diffusion term: A short calculation gives for \(\sigma =K|L\) and \(x_{\nu K L}:=v_{\nu L}-v_{\nu K}\)
with
Using Lemma 7, inequality (41) and the following auxiliary calculation (with \(x_{\nu L}=(v_{\nu L}+\kappa )^+\) and \(x_{\nu K}=(v_{\nu K}+\kappa )^+\))
we can estimate the term \(A\) from above by
Together with Lemma 7, inequality (43) and the auxiliary calculation
we can bound the term \(B\) by
Therefore we can bound \(S_2\) with some constant \(\underline{c}_{D \overline{u}}=\underline{c}_{D}\underline{c}_{\overline{u}}\) by
The last term in the first line of the above inequalities can be neglected, since we only need the first \(H^1\)-seminorm in the following estimate. In the last line we extend the \(H^1\)-seminorm to the full \(H^1\)-norm and use \(1/2\le 1-1/p< 1\) and \(0< (p-1)/p^2\le 1/4\) for \(p\in [2,\infty )\).
Reaction terms: The reaction terms multiplied by the test function can be written as
Using the \(L^\infty \) bounds of Theorem 3 we deduce for \(\alpha _\nu >\beta _\nu \) that
A corresponding estimate holds for \(\alpha _\nu <\beta _\nu \) and therefore we get with \(C_2>0\)
Moser iteration: From \(S_1=S_2+S_3\) together with \(x^{p-1}\le x^{p}+1\) for \(x\ge 0\) and \(w_{\nu ,h}=z_{\nu ,h}^{p/2}\) for \(p\ge 2\) we conclude with some constant \(C_3>0\)
Using Gagliardo–Nirenberg’s inequality (35) and Young’s inequality with \(q=2\), \(q^{\prime }=2\) we can estimate
and the estimate of \(S_4\) can be continued by
with
The constant \(\epsilon \) is fixed such that \(g_1(\epsilon )=0\) holds, i.e.
Setting \(C_4:=C_3 \max (g_2(\epsilon ), 1)\) we find
Now we proceed in a similar way as in the proof of Theorem 3. We set
Moreover we set \(p=2^k\) for \(k\ge 1\). Together with
where \(C_5>0\), we find for all \(k\ge 1\) the recursion formula
Applying (26) we conclude \(b_k\le (4^{4} C_5 b_0)^{2^k}\) and
This estimate holds true for all times \(t_N\), \(N\in \mathbb {N}\). The term \(b_0\) is bounded by Lemma 4. Passing to the limit \(k\rightarrow \infty \) we obtain
The procedure can be done for \(\nu =1,\ldots ,m\) and the result of the theorem follows with \(v_{\nu ,h}^{-}=v_{\nu ,h}^{+}+\kappa -(v_{\nu ,h}+\kappa ) \le v_{\nu ,h}^{+} + \kappa +z_{\nu ,h}\) and the bound of \(v_{\nu ,h}^{+}\) from Theorem 3. \(\square \)
3.7 Comments on 3D
Using the new results concerning the discrete Sobolev–Poincaré inequality in [1], see (38), instead of [26, Theorem 2.2], the uniform exponential decay of the discrete free energy for Voronoi discretized reaction–diffusion systems can be established up to a reaction order less or equal to \(3\) (see Appendix (38) \(p=6\) is allowed now, and see [24, Remark 4.4, p. 2173]).
As in the continuous situation [19], our technique to get uniform upper and lower bounds works also in three space dimensions, provided that the order of the reaction source terms (see (5)) is less or equal to \(\frac{5}{3}\), see Lemma 5.
4 Numerical example
In order to illustrate the discretization approach, we consider the Michaelis-Menten-Henri reaction mechanism [3], present in the modeling of a pattern doubling process by a catalytic cross-linking of a spacer, triggered by residual acid diffusion from a previously developed primary structure into the spacer, see [17]. Double patterning is a possible option used in optical lithography to create lines with a structure size less then 48 nm, see [10, 34, 35].
The Michaelis–Menten–Henri reaction mechanism (MMH) [3] is well known in biology and chemistry, and can be written symbolically as
where \(\mathsf X_1\) is a precursor, \(\mathsf X_2\) is an acid, \(\mathsf X_3\) is an intermediate, and \(\mathsf X_4\) is a cross-linked polymer. Correspondingly, the set \(\mathcal {R}\) consists of two pairs of vectors, namely \(\varvec{\alpha }_1=(1,1,0,0)\) and \(\varvec{\beta }_1=(0,0,1,0)\) for the first reaction as well as \(\varvec{\alpha }_2=(0,0,1,0)\) and \(\varvec{\beta }_2=(0,1,0,1)\) for the second reaction. Due to the definition of the reaction term (3), the net production rates of the species are given by
The stoichiometric subspace \(\mathcal {S}\) and its orthogonal complement \(\mathcal {S}^{\bot }\) are spanned by
Every element of \(\mathcal {S}^{\bot }\) creates one invariant of the system, i.e., \(R_1+R_3+R_4\equiv 0\) and \(R_2+R_3\equiv 0\) hold, and hence
are conserved during the time evolution, exploiting the homogeneous Neumann boundary conditions and integration over the entire domain. At thermodynamic equilibrium the chemical activities are constant over all control volumes \(K\in \mathcal {V}\), therefore the solution can be obtained by solving
together with (32), (33). In the model that we have in mind [16, 17], the diffusion coefficient of \(\mathsf X_2\) is given by
where \(\varphi _1\), \(\varphi _2\), \(\varphi _3\) are so-called lumped constants, see [17]. All other diffusion coefficients are assumed to be piecewise constant and independent of the concentration. Time integration is done using the fully implicit Euler discretization, where the initial guess of the Newton iteration is predicted by a linear extrapolation in time of the last two previous accepted solutions for the chemical potentials. The time step of the method is adapted to the variation of the free energy, the dissipation rate of the system, and the number of necessary iterations for Newton’s method.
The stopping criterion for Newton’s method is \(\mathord {\left||\varvec{\delta }_{v}\right||}_{L^{\infty }}\le \epsilon _r\), where \(\varvec{\delta }_{v}\) is the Newton update of chemical potentials and \(\epsilon _r\) is a given tolerance [21]. The resulting linear systems are solved by the sparse direct solver Pardiso [42, 43]. Due to roundoff errors the invariants could be driven away during the time evolution. This is a known behavior in the context of solving ODEs with linear constraints numerically, see [8]. Therefore we introduce two Lagrange multipliers to stay in the affine subspace \(\varvec{U}+\widehat{\mathcal {U}}\), see (12) and (32), (33), and use the relative mass invariant error as additional stopping criterion.
We consider a heterostructure consisting of two different materials. The interface between different materials is aligned to the Delaunay triangulation that is dual to the Voronoi grid. One half of the used mesh is depicted in Fig. 2. We note that instead of the Voronoi boxes the dual Delaunay triangulation is shown. The materials are represented by different colors in the mesh and the used parameters are collected in Table 1. At thermodynamic equilibrium, the chemical activities are given by
The simulation parameters in the blue region are chosen in such a way that the catalyst \(\mathsf X_2\) is created by the intermediate \(\mathsf X_3\). In the red and green region the forward direction of the MMH mechanism dominates. We expect that the catalyst \(\mathsf X_2\) from the blue region diffuses and creates together with \(\mathsf X_1\) the intermediate \(\mathsf X_3\) which quickly degrades into \(\mathsf X_4\) by releasing \(\mathsf X_2\). On the slow timescales the created structures dissolve by diffusion and the thermodynamic equilibrium solution is attained. Technologically, the thickness of the reaction front of \(\mathsf X_4\) can be controlled by an additional fast reaction which neutralizes the catalyst \(\mathsf X_2\), therefore, the MMH reaction stops, see [17]. But this part of the technological process is not included in the example. Figure 3 shows the time evolution of the free energy, the dissipation rate divided in the caused by reaction and diffusion, and the relative error of the two invariants. Every time scale in the system represents one plateau in the curve of the dissipation rate. The reaction part and the diffusion part provide information whether the system is reaction-dominated or diffusion-dominated, i.e. in the first interval from \(10^{-30}\) to \(10^{-8}\) the system is reaction dominated. Then energy of the system disappears by the reaction and diffusion. After the whole region is filled with \(\mathsf X_4\) the system is diffusion dominated. The oscillating behavior of the dissipation rate curves correlates with the number of gridlines in the refined region. In this region, the front has to pass over the same distance, in order to jump to the next grid points. Therefore the period of the oscillations is larger than in the non-refined region.
In Fig. 4 the time evolution of the four species concentrations is shown at different time steps. Every column in Fig. 4 is represented by a straight (light gray) line in Fig. 3. Up to \(T=5\cdot 10^{-3}\), the species \(\mathsf X_3\) decomposes into \(\mathsf X_2\) and \(\mathsf X_4\). Now the reaction of \(\mathsf X_2\) and \(\mathsf X_1\) starts and the profile of \(\mathsf X_4\) grows (\(T=25\) and \(T=250\)). Since the lifetime of \(\mathsf X_3\) is very short, the concentration of \(\mathsf X_3\) is very small. Once \(\mathsf X_1\) is fully converted into \(\mathsf X_4\), the dissipation rate of the reactions drops down and the slow diffusion and the backward reactions start (\(T=10^{13}\)) and converges to the thermodynamic equilibrium solution (last column).
We emphasize that our method can reach—up to machine epsilon—the thermodynamic equilibrium (approx. at \(T=10^{20}\)) and shows the expected monotonous (and exponential) decay of the free energy and the expected non-negativity of the dissipation rate, see Lemma 2. Moreover the relative error of the two invariants (32) and (33) is of the order of machine epsilon over all \(50\) orders of magnitude in time. We needed \(9{,}289\) time steps and \(49{,}890\) Newton steps. The Newton method shows almost quadratic convergence rates and passes the convergence criteria in \(3\) or \(4\) steps. During the marching of the reaction front the Newton iteration needs \(5\) or \(6\) steps, in order to pass the convergence criteria. Since the reaction front can not be accurately predicted by the linear extrapolation predictor, the initial guess of the Newton method is not in the region of quadratic convergence leading to a larger number of necessary Newton steps. To overcome this problem, one would need a more accurate prediction of the reaction front, which would be possible to implement, but was not in the focus of this contribution.
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Acknowledgments
A. Fiebach was supported by the European Commission within the Seventh Framework Programme \((FP7)\) MD3 “Material Development for Double Exposure and Double Patterning”. A. Linke and A. Glitzky are partially supported by the DFG Research Center Matheon Mathematics for key technologies. The authors would like to thank Klaus Gärtner, Jens Griepentrog, and Hang Si for their assistance and for valuable discussions.
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Appendices
Appendix A: Discrete Gagliardo–Nirenberg inequality
In the following, we recapitulate from [1] a discrete version of the Gagliardo–Nirenberg inequality [40] and a discrete Sobolev–Poincaré inequality on admissible finite volume meshes [12]. The proof uses functions of bounded variation and is done under the general assumption on the mesh, that
see [1, Eq. (4)]. In the case of Voronoi meshes this condition holds true with \(\xi =\frac{1}{2}\).
Lemma 5
(see [1, Th. 3 and Th. 4]) Let \(\varOmega \) be an open bounded polyhedral domain of \(\mathbb {R}^d\), \(d\ge 2\). Let \(\mathcal {M}=(\mathcal {P},\mathcal {V},\mathcal {E})\) be a given admissible finite volume mesh which satisfies (34).
-
Then, there exists a constant \(C>0\) only depending on \(d\) and \(\varOmega \) (but not on \(\mathcal {M}\)) such that
$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}\le C_{gn,p}\mathord {\left||w_h\right||}_{L^1}^{1-\theta }\mathord {\left||w_h\right||}_{H^1,\mathcal {M}}^{\theta },\quad C_{gn,p}\!:=\!\frac{C}{\xi ^{\theta /2}}, \quad \forall w_h\!\in \! X_{\mathcal {V}}(\mathcal {M}),\qquad \end{aligned}$$(35)where
$$\begin{aligned} \theta =\frac{2d(1-p)}{(d+2)p},\quad p\in {\left\{ \begin{array}{ll} \left[ 1,\infty \right) &{} \text { if } d=2,\\ \left[ 1,2d/(d-2)\right] &{} \text { if } d>2. \end{array}\right. } \end{aligned}$$(36) -
Let \(d\le 2\). Then for all \(p\in [1,\infty )\) there exists a constant \(C>0\) only depending on \(p\), \(d\) and \(\varOmega \) (but not on \(\mathcal {M}\)) such that
$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}\le \frac{C}{\xi ^{1/2}}\mathord {\left||w_h\right||}_{H^1,\mathcal {M}} \quad \forall w_h\in X_{\mathcal {V}}(\mathcal {M}). \end{aligned}$$(37) -
Let \(d> 2\). Then for all \(1\le p \le \frac{2d}{d-2}\) there exists a constant \(C>0\) only depending on \(p\), \(d\) and \(\varOmega \) (but not on \(\mathcal {M}\)) such that
$$\begin{aligned} \mathord {\left||w_h\right||}_{L^p}\le \frac{C}{\xi ^{1/2}}\mathord {\left||w_h\right||}_{H^1,\mathcal {M}} \quad \forall w_h\in X_{\mathcal {V}}(\mathcal {M}). \end{aligned}$$(38)
Remark 8
An inspection of the proof shows that \(C_{gn,p}\) depends continuously on \(p\) and can therefore be estimated uniformly for \(p\in [2,3]\). The results of [1, Th. 3 and Th. 4] allow us to handle all Voronoi meshes, including arbitrarily anisotropic grids. We remark that in the case of homogeneous Dirichlet boundary conditions a discrete Sobolev–Poincaré inequality on more general meshes than admissible meshes is proven in [13, Sec. 5].
In the proof of Theorem 3 we need a special case of the discrete Gagliardo–Nirenberg inequality:
Corollary 2
Let \(\varOmega \) be an open bounded polygonal domain in \(\mathbb {R}^2\). Let \(\mathcal {M}=(\mathcal {P},\mathcal {V},\mathcal {E})\) be a given admissible finite volume mesh which satisfies (34). For any \(\epsilon >0\) and any \(p\in (1,3]\) there exists a constant \(c_{\epsilon ,p}>0\) such that
(For \(\epsilon \rightarrow 0\) it follows \(c_{\epsilon ,p}\rightarrow \infty \)).
Proof
For \(N>1\) we define the function
Adding and subtracting \(\chi (w_h)\) we obtain
The first term of (40) can be estimated by
Applying (35) for the second term of (40) we obtain
We estimate
and since \(\mathord {\left|\frac{\chi (w_L)-\chi (w_K)}{w_L-w_K}\right|}\le 2\) and \(\mathord {\left|\chi (w_h)\right|}\le \mathord {\left|w_h\right|}\) we deduce
All together we find
with \(c_{\epsilon ,p}=c_p(2 N)^{p-1}\) and \(\epsilon =\frac{c_p C^p 2^{p-1}}{\ln N}\). \(\square \)
Appendix B: Technical lemmas
In this part we collect some auxiliary results, which we use in Sect. 3.4 and in Sect. 3.6.
Lemma 6
Let \(x,y,p\in \mathbb {R}\), \(x,y>0\).
-
1.
For \(p\ge 2\) the following inequalities hold:
$$\begin{aligned} \frac{4(p-1)}{p^2}\left( x^{p/2}-y^{p/2}\right) ^2 \le (x-y)(x^{p-1}-y^{p-1})\le \left( x^{p/2}-y^{p/2}\right) ^2.\qquad \end{aligned}$$(41) -
2.
For \(p\ge 1\), we have
$$\begin{aligned} \frac{1}{p}(x^{p}-y^{p}) \le x^{p-1}(x-y). \end{aligned}$$(42) -
3.
Finally, for \(p\ge 2\) the inequalities
$$\begin{aligned} \frac{2}{p^2}(x^{p/2}-y^{p/2})^2\le (x^{p-2}+y^{p-2})(x-y)^2\le 2(x^{p/2}-y^{p/2})^2 \end{aligned}$$(43)are fulfilled.
Proof
-
1.
For \(z\ge 1\), we consider the function
$$\begin{aligned} f(z)=(z-1)(z^{p-1}-1)-\frac{4(p-1)}{p^2}(z^{p/2}-1)^2. \end{aligned}$$The first and second derivatives of \(f\) are given by
$$\begin{aligned} \frac{d}{d z} f(z)&= \frac{(p-2)^2}{p} z^{p-1} - (p-1) z^{p-2} +\frac{4(p-1)}{p}z^{p/2-1} -1,\\ \frac{d^2}{d z^2} f(z)&= \frac{(p-2)(p-1)}{p} \left( ((p-2)z-p)z^{p-3}+2z^{p/2-2}\right) . \end{aligned}$$It is easy to see that \(f(1)=0\), \(f^{\prime }(1)=0\) and \(f^{\prime \prime }(1)=0\). Further, we deduce from \(f^{\prime \prime }(z)>0\) for \(z>1\) that \(f^{\prime }(z)>0\) and \(f(z)>0\). With \(z=x/y\), \(x\ge y > 0\) we find
$$\begin{aligned} 0\le f(x/y)= (x-y)(x^{p-1}-y^{p-1}) -\frac{4(p-1)}{p^2}\left( x^{p/2}-y^{p/2}\right) ^2 \end{aligned}$$and finally by using Muirhead’s inequality
$$\begin{aligned} \left( x^{p/2}-y^{p/2}\right) ^2-(x-y)(x^{p-1}-y^{p-1})= x^{p-1}y+x y^{p-1}-2 x^{p/2} y^{p/2} \ge 0 \end{aligned}$$holds. The case \(y=0\) is trivial.
-
2.
For the second statement we consider for \(z\ge 1\) the function
$$\begin{aligned} f(z)=\frac{p-1}{p} z^{p}- z^{p-1}+ \frac{1}{p}. \end{aligned}$$Since \(f(1)=0\), the first derivative of \(f\)
$$\begin{aligned} \frac{d}{d z} f(z) = (p-1) z^{p-2}(z-1)\ge 0 \end{aligned}$$implies \(f(z)\ge 0\). Setting \(z=x/y\), \(x\ge y > 0\) we find
$$\begin{aligned} 0 \le y^p f(x/y)= \left( \frac{p-1}{p}\frac{x^p}{y^p} -\frac{x^{p-1}}{y^{p-1}} +\frac{1}{p}\right) y^p \!=\!x^{p-1}(x-y)-\frac{1}{p}(x^p-y^p). \end{aligned}$$For \(x\le y\) it results
$$\begin{aligned} \frac{1}{p}(x^p-y^p)\le (x^p-y^p) \le x^{p-1}(x-y x^{-p+1}) \le x^{p-1}(x-y). \end{aligned}$$ -
3.
Now let
$$\begin{aligned} f(z)=(z^{p-2}+1)(z-1)^2-\frac{2}{p^2}(z^{p/2}-1)^2. \end{aligned}$$The first and second derivatives are given by
$$\begin{aligned} \frac{d}{d z} f(z)&= 2(z-1)(z^{p-2}+1)+(p-2)(z-1)^2 z^{p-3} -\frac{2}{p}(z^{p/2}-1)z^{p/2-1},\\ \frac{d^2}{d z^2} f(z)&= \frac{1}{p}\left( 2p+(p-2)z^{p/2-2}+z^{p-4} g(z)\right) \end{aligned}$$with
$$\begin{aligned} g(z)&= (p-3)(p-2)p-2(p-2)(p-1)p z+(p-1)(p^2-2)z^2,\\ g^{\prime }(z)&= 2(p-1)\left( (p^2-2) z-(p-2) p\right) ,\\ g^{\prime \prime }(z)&= 2(p-1)(p^2-2). \end{aligned}$$From \(g^{\prime \prime }(z)>0\) for \(z>1\) and \(p\ge 2\) we see that \(g(z)\) is a convex function. Furthermore \(f(z)\) is convex since \(g^{\prime }(1)=4(p-1)^2>0\), \(g(1)=2\) and \(f^{\prime \prime }(z)>0\). Using \(f^{\prime }(1)=f(1)=0\) we get \(f(z)>0\) and with \(z=x/y\), \(x\ge y > 0\) the first inequality of (43). The last assertion follows from
$$\begin{aligned}&(x^{p/2}-y^{p/2})^2-\frac{1}{2}(x^{p-2}+y^{p-2})(x-y)^2\\&\quad = \frac{1}{2}\left( x^p+y^p-x^{p-2}y^2-x^2 y^{p-2}\right) +\left( x^{p-1}y+xy^{p-1}-2x^{p/2}y^{p/2}\right) \end{aligned}$$together with Muirhead’s inequality for the term
$$\begin{aligned} x^p+y^p&\ge x^{p-2}y^2+x^2y^{p-2},&x^{p-1}y+xy^{p-1}&\ge 2 x^{p/2}y^{p/2}. \end{aligned}$$
\(\square \)
Lemma 7
Let \(x\) be a real number. Then
hold. We define the value of the functions at \(x=0\) as limit \(x\rightarrow 0\).
Proof
Using the power series of \(\sinh (x)\) and \(\cosh (x)\) we obtain
\(\square \)
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Fiebach, A., Glitzky, A. & Linke, A. Uniform global bounds for solutions of an implicit Voronoi finite volume method for reaction–diffusion problems. Numer. Math. 128, 31–72 (2014). https://doi.org/10.1007/s00211-014-0604-6
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DOI: https://doi.org/10.1007/s00211-014-0604-6