Weierstraß-Institut. im Forschungsverbund Berlin e.v. Preprint ISSN

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1 Weierstraß-Institut für Angewandte Analysis und Stochastik im Forschungsverbund Berlin e.v. Preprint ISSN Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme Olaf Klein, Peter Philip submitted: September 26, 2003 Weierstrass Institute for Applied Analysis and Stochastics Mohrenstraße 39 D Berlin Germany klein@wias-berlin.de philip@wias-berlin.de No. 871 Berlin 2003 W I A S 2000 Mathematics Subject Classification. 45K05, 65M99, 35K05, 35K55, 65N22, 47H10, 80A20. Key words and phrases. Integro-partial differential equations. Finite volume method. Nonlinear parabolic PDEs. Integral operators. Nonlocal interface conditions. Diffuse-gray radiation. Maximum principle. This work has been supported by the DFG Research Center Mathematics for key technologies FZT 86) in Berlin and by the German Federal Ministry for Education and Research BMBF) within the program Neue Mathematische Verfahren in Industrie und Dienstleistungen New Mathematical Methods in Manufacturing and Service Industry ) # 03SPM3B5.

2 Edited by Weierstraß-Institut für Angewandte Analysis und Stochastik WIAS) Mohrenstraße 39 D Berlin Germany Fax: preprint@wias-berlin.de World Wide Web:

3 Abstract This article presents a finite volume scheme for transient nonlinear heat transport equations coupled by nonlocal interface conditions modeling diffusegray radiation between the surfaces of both open and closed) cavities. The model is considered in three space dimensions; modifications for the axisymmetric case are indicated. Proving a maximum principle as well as existence and uniqueness for roots to a class of discrete nonlinear operators that can be decomposed into a scalar-dependent sufficiently increasing part and a benign rest, we establish a discrete maximum principle for the finite volume scheme, yielding discrete L -L a priori bounds as well as a unique discrete solution to the finite volume scheme. 1 Introduction Modeling and numerical simulation of conductive-radiative heat transfer has become a standard tool to support and improve numerous industrial processes such as crystal growth by the Czochralski method and by the physical vapor transport method see [DNR + 90] and [KPSW01], respectively) to mention just two examples. The physical modeling of conductive-radiative heat transfer is well-understood see e.g. [SC78], [Mod93]), and, for models of diffuse-gray radiation, a mathematical theory of existence and uniqueness of weak solutions has been developed in recent years see [LT01] and references therein). Mathematical treatments of discretization methods in the context of conductive-radiative heat transfer are still scarce in the literature, especially, if one is interested in nonconvex domains containing nonconvex cavities. For such a general situation, the authors are only aware of [Tii98], where a finite element approximation is considered for a stationary conductive-radiative heat transfer problem. Mathematical research on the finite volume method has been very active in recent years see [EGH00] for an extensive survey). However, to the authors knowledge, a finite volume discretization of the equations governing conductive-radiative heat transfer has not yet been studied in a mathematical context, even though, as e.g. shown by the numerical results in [KPSW01] and [KP03], it has been used to develop efficient and accurate codes for numerical simulations. The purpose of this article is to derive and analyze a finite volume discretization of transient heat equations coupled by nonlocal operators modeling diffuse-gray radiation between surfaces of cavities within a rigorous mathematical framework. The general setting is somewhat similar to [Tii98], however, in contrast to [Tii98], 1

4 in the present article, transient heat transport is treated, and heat conduction is also considered inside closed cavities, with a jumping diffusion coefficient at the interface. Moreover, the emissivity is allowed to depend on the temperature. The finite volume scheme leads to a nonlinear and nonlocal system of equations, the solvability of which is not at all obvious. The proof of existence and uniqueness of a discrete solution is based on a maximum principle for the discrete nonlinear operator as well as on monotonicty and regularity considerations. The maximum principle, existence and uniqueness are first established for roots to a class of continuous discrete nonlinear operators H, where it is assumed that the components H i of H can be decomposed into sufficiently increasing scalar-dependent continuous functions b i and h i, and a Lipschitz continuous vector-dependent function g i such that g i h i satisfy a boundedness condition s. Th. 4.2). Further research concerning the convergence of the scheme and corresponding error estimates is currently under way. The paper is organized as follows: In Sec. 2, the governing equations of transient conductive heat transfer are stated, completed by nonlocal interface and boundary conditions arising from the modeling of diffuse-gray radiation. Section 2 also provides the precise mathematical setting. The discrete scheme is developed in Sec. 3, where the nonlocal radiation operators are discretized in 3.3, also providing some important properties of the resulting discrete nonlocal operators. Section 3.6 discusses modifications occurring in the axisymmetric case. The proof of existence and uniqueness of a discrete solution to the finite volume scheme is the subject of Sec. 4, where the root problem is solved in 4.1, and the finite volume scheme is considered in 4.2. The main result is presented in Th Transient Heat Transport Including Conduction and Diffuse-Gray Radiation 2.1 Transient Heat Equations Transient conductive-radiative heat transport is considered on a time-space cylinder [0, T ] Ω, where: A-1) T R +, Ω = Ω s Ω g, Ω s Ω g =, and each of the sets Ω, Ω s, Ω g, is a nonvoid, polyhedral, bounded, and open subset of R 3. The set Ω s represents the domain of a solid apparatus enclosing gas cavities represented by Ω g. That Ω g is enclosed by Ω s means see Fig. 1): A-2) Ω s = Ω Ω g, where denotes a disjoint union. Thus, Σ := Ω g = Ω s Ω g, and Ω = Ω s \ Σ. 2

5 Heat conduction is considered throughout Ω. Nonlocal radiative heat transport is considered between points on the surface Σ of Ω g as well as between points on the surfaces of open cavities such as O 1 and O 2 in Fig. 1). However, to avoid introducing additional boundary conditions, open cavities are not part of Ω, i.e. heat conduction is not considered in open cavities see Sec. 2.3 below for details). Ω s Ω Σ := Ω g O 1 Ω g O 2 Ω s = Ω Σ Σ = Ω s Ω g Ω s Ω = Ω s \ Σ Figure 1: Possible shape of a 2-dimensional section through the 3-dimensional domain Ω = Ω s Ω g with open cavities O 1 and O 2. Note that, according to A-2), Ω g is engulfed by Ω s, which can not be seen in the 2-dimensional section. Transient heat conduction is described by ε m θ) t divκ m θ) = f m t, x) in Ω m m {s, g}), 2.1) where θt, x) R + 0 represents absolute temperature, depending on the time coordinate t and on the space coordinate x; the continuous, strictly increasing, nonnegative functions ε m CR + 0, R + 0 ) represent the internal energy in the solid and in the gas, respectively, κ m R + 0 represent the thermal conductivity in solid and gas, respectively, assumed constant for simplicity, and f m is a heat source due to some heating mechanism. In practice, for many heating mechanisms such as induction or resistance heating, one has f g = 0. Throughout this paper, A-3) A-5) are assumed, where: A-3) For m {s, g}, ε m : R + 0 R + 0 is continuous and at least of linear growth, i.e. there is C ε R + such that A-4) For m {s, g}: κ m R + 0. ε m θ 2 ) θ 2 θ 1 ) C ε + ε m θ 1 ) θ 2 θ 1 0). A-5) For m {s, g}: f m L 0, T, L Ω m )), f m 0 a.e. 3

6 2.2 Nonlocal Interface Conditions For simplicity, the temperature is assumed to be continuous at the interface Σ: θt, ) Ωs = θt, ) Ωg on Σ t [0, T ]), 2.2) where denotes restriction. Continuity of the heat flux on the interface between solid and gas, where one needs to account for radiosity R and for irradiation J, yields the following interface condition, coupling the two equations in 2.1) m {s, g}): κ g θ) Ωg n g + Rθ) Jθ) = κ s θ) Ωs n g on Σ. 2.3) Here, denotes the scalar product, and n g denotes the unit normal vector pointing from gas to solid. It is assumed that the solid is opaque, and Rθ) and Jθ) are computed according to the net radiation model for diffuse-gray surfaces, i.e. reflection and emittance are taken to be independent of the angle of incidence and independent of the wavelength. At each point of the surface Σ of the gas cavity, the radiosity is the sum of the emitted radiation Eθ) and of the reflected radiation J r θ): According to the Stefan-Boltzmann law, where it is assumed that A-6) σ R +, ɛ : R + 0 ]0, 1] is continuous. R = E + J r. 2.4) Eθ) = σ ɛθ) θ 4, 2.5) Here, σ represents the Boltzmann radiation constant, and ɛ represents the potentially temperature-dependent emissivity of the solid surface. Using the presumed opaqueness together with Kirchhoff s law yields Due to diffuseness, the irradiation can be calculated as J r = 1 ɛ) J. 2.6) Jθ) = KRθ)) 2.7) using the integral operator K defined by Kρ)x) := Λx, y) ωx, y) ρy) dy a.e. x Σ), 2.8) Σ where the visibility factor Λx, y) is 1 or 0, depending on the points x and y being mutually visible or not. The view factor ω is defined almost everywhere by ng y) x y) ) n g x) y x) ) ωx, y) := π y x) y x) ) 2 a.e. x, y) Σ 2, x y ). 2.9) 4

7 According to [Tii97b, Lem. 2], K is a positive compact operator from L p Σ) into itself for each p [1, ], and, since Σ forms an enclosure, K = 1. Moreover, for the closed surface Σ, the following holds conservation of radiation energy, [Tii97a, Lem. 1]): Λx, y) ωx, y) dy = 1 a.e. x Σ). 2.10) Σ Combining 2.4) through 2.7) provides the following nonlocal equation for the radiosity Rθ): Rθ) 1 ɛθ) ) KRθ)) = σ ɛθ) θ ) One can write 2.11) in the form where the operator G θ is defined by G θ Rθ)) = Eθ), 2.12) G θ ρ) := ρ 1 ɛθ) ) Kρ). 2.13) For the following Lem. 2.1, it is not necessary to assume any regularity of ɛ and θ: Lemma 2.1. If the functions ɛ : R + 0 ]0, 1] and θ : Σ R + 0 are measurable, then, for each p [1, ], the operator G θ maps L p Σ) into itself and has a positive inverse. Proof. Since the function ɛ θ is a measurable function with values in ]0, 1], the lemma follows from [LT01, Lem. 2], where our ɛ θ plays the role of ɛ in [LT01, Lem. 2]. Lemma 2.1 allows to state 2.12) as From 2.11) and 2.7), it is such that 2.3) becomes Rθ) = G 1 θ Eθ)). 2.14) Rθ) Jθ) = ɛθ) KRθ)) σ θ 4), 2.15) κ g θ) Ωg n g ɛθ) KRθ)) σ θ 4) = κ s θ) Ωs n g on Σ. 2.16) 2.3 Nonlocal Outer Boundary Conditions Definition 2.2. A family A i ) i I of subsets of R d is called a partition of A R d iff with respect to the relative topology on A) A = i I A i and int A i int A j = for each i j. Thus, in the sense of Def. 2.2, Ω s, Ω g ) is a partition of Ω. 5

8 Definition and Remark 2.3. Let convω) denote the closed convex hull of Ω, and define O := intconvω)) \ Ω, Γ Ω := Ω O, Γ := O, and Γ ph := convω) O. Then Γ Ω, Γ ph ) forms a partition of Γ. The set O is the domain of the open radiation region e.g., one has O = O 1 O 2 in Figures 1 and 2). On the interface Γ Ω between Ω and the open radiation region O, one has κ s θ n s + R Γ θ) J Γ θ) = 0 on Γ Ω 2.17) in analogy with 2.3), where n s is the outer unit normal vector to the solid. To allow for radiative interactions between surfaces of open cavities and the ambient environment, including reflections at the cavity s surfaces, the set Γ ph as defined above, is used as a black body phantom closure see Fig. 2), emitting radiation at an external temperature θ ext, A-7) θ ext R + 0. Thus, ɛ 1 on Γ ph, leading to R Γ θ)x) = σ θ 4 ext x Γ ph ). 2.18) Here and in the following, it is assumed that the apparatus is exposed to a black body environment e.g. a large isothermal room) radiating at θ ext. A relation analogous to 2.15) holds on Γ Ω, and using it in 2.17) yields κ s θ n s ɛθ) K Γ R Γ θ)) σ θ 4) = 0 on Γ Ω, 2.19) where K Γ is defined analogous to K in 2.8), except that the integration is carried out over Γ instead of over Σ. Ω s Ω = Ω s Ω g Ω g Σ := Ω g = Ω s Ω g O 1 Ω g O 2 O := intconvω)) \ Ω = O 1 O 2 Γ Ω := Ω O Γ ph := convω) O Ω s Γ := O = Γ Ω Γ ph Figure 2: For the domain of Fig. 1, the surfaces of radiation regions are shown. The open radiation regions O 1 and O 2 are artificially closed by the phantom closure Γ ph. As in Fig. 1, Fig. 2 depicts a 2-dimensional section through the 3-dimensional domain. On parts of Ω that do not interact radiatively with other parts of the apparatus, i.e. on Ω \ Γ Ω, the Stefan-Boltzmann law provides the outer boundary condition κ s θ n s σ ɛθ) θ 4 ext θ 4 ) = 0 on Ω \ Γ Ω. 2.20) 6

9 2.4 Initial Condition The initial condition reads θ0, x) = θ init x), x Ω, where it is assumed that A-8) θ init L Ω, R + 0 ). 3 The Discrete Scheme We assume A-1) A-8) throughout this section. 3.1 Discretization of Time and Space Domain A discretization of the time domain [0, T ] is given by an increasing finite sequence 0 = t 0 < < t N = T, N N. The notation k ν := t ν t ν 1 will be used for the time steps. An admissible discretization of the space domain Ω is given by a finite family T := ω i ) i I of subsets of Ω satisfying a number of assumptions, subseqently denoted by DA- ). DA-1) T = ω i ) i I forms a partition of Ω according to Def. 2.2, and, for each i I, ω i is a nonvoid, polyhedral, connected, and open subset of Ω. From T, one can define discretizations of Ω s and Ω g : For m {s, g} and i I, let ω m,i := ω i Ω m, I m := { j I : ω m,j }, T m := ω m,i ) i Im. 3.1) To allow the incorporation of the interface condition 2.16) into the scheme see 3.3a) and 3.7b) below), it is assumed that, if some ω i has a 2-dimensional intersection with the interface Σ, then it lies on both sides of the intersection. More precisely: DA-2) For each i I: reg ω s,i Σ = reg ω g,i Σ, where reg denotes the regular boundary of a polyhedral set, i.e. the parts of the boundary, where a unique outer unit normal vector exists see Fig. 3), reg :=. Integrating 2.1) over [t ν 1, t ν ] ω m,i, applying the Gauss-Green integration theorem, and using implicit time discretization yields kν 1 εm θ ν ) ε m θ ν 1 ) ) tν κ m θ ν n ωm,i = kν 1 f m, 3.2) ω m,i ω m,i ω m,i where θ ν := θt ν, ), and n ωm,i denotes the outer unit normal vector to ω m,i. The time discretization of the interface and boundary conditions 2.16), 2.19), and 2.20), respectively, is also done implicitly, except for the temperature dependence of 7 t ν 1

10 Ω s ω s,1 ω 2 ω 3 Σ ω g,1 Ω g Ω s Figure 3: Illustration of condition DA-2): Ω s consists of the outer wall of the box as well as of the region above the gray horizontal plane, which is contained in Σ; Ω g consists of the region below that plane and engulfed by the wall. Both ω 1 and ω 2 satisfy DA-2) where reg ω s,2 Σ = = ω g,2 ), however ω 3 does not satisfy DA-2) reg ω s,3 Σ = ω g,3 ). the emissivity, which is discretized explicitly, thereby, e.g., substantially simplifying the use of Newton s method for the nonlinear solver. More precisely, the approximation Rθ ν 1, θ ν ) of the radiosity Rθ) is supposed to satisfy a discretized version of 2.11), where ɛθ) is replaced by ɛθ ν 1 ), and θ 4 is replaced by θ 4 ν also cf. 3.15) below). Analogously, R Γ θ) is replaced by an approximation R Γ θ ν 1, θ ν ). The time discretizations of 2.16), 2.19), and 2.20) thus read κ g θ ν ) Ωg n g ɛθ ν 1 ) KRθ ν 1, θ ν )) σ θ 4 ν) = κs θ ν ) Ωs n g on Σ, 3.3a) and respectively. κ s θ ν n s ɛθ ν 1 ) K Γ R Γ θ ν 1, θ ν )) σ θ 4 ν) = 0 on ΓΩ, 3.3b) κ s θ ν n s σ ɛθ ν 1 ) θ 4 ext θ 4 ν) = 0 on Ω \ Γ Ω, 3.3c) 3.2 Approximation of Space Integrals, Interface and Boundary Conditions The finite volume scheme is furnished by using the time-discrete interface and boundary conditions 3.3) in 3.2) and by approximating integrals by quadrature formulas. To approximate θ ν by a finite number of discrete unknowns θ ν,i, i I, precisely one value θ ν,i is associated with each control volume ω i. Introducing a discretization point x i ω i for each control volume ω i, the θ ν,i can be interpreted as θ ν x i ) cf. [FL01]). Moreover, the discretization makes use of regularity assumptions concerning the partition ω i ) i I that can be expressed in terms of the x i see DA-3), DA-4), and DA-5) below). 8

11 The first integral in 3.2) is approximated by ω m,i εm θ ν ) ε m θ ν 1 ) ) ε m θ ν,i ) ε m θ ν 1,i ) ) λ 3 ω m,i ), 3.4) where, here and in the following, λ d, d {2, 3}, denotes d-dimensional Lebesgue measure. Approximation 3.4) is exact if θ ν and θ ν 1 are constant inside ω m,i. The boundary of each control volume ω m,i can be decomposed according to see Fig. 4) ω m,i = ) ω m,i Ω m ωm,i Ω ) ω m,i Σ ). 3.5a) Recalling A-1), A-2), and Def. and Rem. 2.3, outer boundary sets are decomposed further into ω s,i Ω = ) ω s,i Γ Ω ωs,i Ω \ Γ Ω ) ), 3.5b) whereas ω g,i Ω =. ω m,i Ω m ω s,1 ω 1 = ω s,1 ω g,2 ωs,2 ω g,2 Ω s ω g,3 ω s,3 ω m,i Ω m ω m,i Ω ω s,1 ω s,2 ω g,2 Ω g ω g,3 ω s,3 ω m,i Ω ω m,i Σ ω g,3 ω s,1 Ω s ω s,3 ωg,2 ω g,3 ω s,3 ω m,i Σ Figure 4: Illustration of the decomposition of the boundary of control volumes ω m,i according to 3.5a). The lower control volume ω 3 is not admissible, as it has 2- dimensional intersections with both Σ and Ω see Rem. 3.1). To guarantee that there is a discretization point x i in each of the integration domains occurring in 3.5), it is assumed that the discretization T respects interfaces and outer boundaries in the following sense: DA-3) For each m {s, g}, i I m : x i ω m,i. In particular, if ω s,i and ω g,i, then x i ω s,i ω g,i. DA-4) For each i I, the following holds: If λ 2 ω i Γ Ω ) 0, then x i ω i Γ Ω ; and, if λ 2 ωi Ω \ Γ Ω ) ) 0, then x i ω i Ω \ Γ Ω cf. Fig. 5). Remark 3.1. Suppose a control volume ω i has a 2-dimensional intersection with both Ω and Σ. Then, by DA-2), ω s,i and ω g,i. Thus, by DA-3), x i Σ. On the other hand, by DA-4), x i Ω, which means that A-2) is violated. It is thus shown that ω i can not have 2-dimensional intersections with both Ω and Σ. In particular, the lower control volume ω 3 in Fig. 4 is not admissible. 9

12 x 1 x 2 x 3 ω m,1 ω m,2 ω m,1 ω m,4 ω m,1 ω m,2 ω m,3 ω m,1 ω m,5 x 4 ω m,4 ω m,5 ω m,7 ω m,6 x 5 x 6 x 7 ω m,7 ω m,3 ω m,7 ω m,4 ω m,7 ω m,6 Figure 5: Illustration of conditions DA-4) with Γ Ω = ) and DA-5) as well as of the partition of ω m,i Ω m according to 3.8). One has nb m 1) = {2, 4, 5} and nb m 7) = {3, 4, 6}. Using the boundary condition 3.3c) leads to the following approximation: κ s θ ν n ωs,i σ ɛθ ν 1,i ) θext 4 θν,i) λ 4 2 ωs,i Ω\Γ Ω ) ). 3.6) ω s,i Ω\Γ Ω ) The nonlocal boundary condition 3.3b) and the nonlocal interface condition 3.3a) yield κ s θ ν n ωs,i = ɛθ ν 1 ) K ) Γ R Γ θ ν 1, θ ν )) σ θν a) ω s,i Γ Ω ω s,i Γ Ω and ω m,i Σ κ m θ ν n ωm,i = ɛθ ν 1 ) KRθ ν 1, θ ν )) σ θν) 4, 3.7b) ω i Σ respectively. However, the approximation of the nonlocal terms K Γ R Γ θ ν 1, θ ν )) and KRθ ν 1, θ ν )) is more involved and is the subject of Sec. 3.3 below. To approximate the integrals over ω m,i Ω m, this set is partitioned further see Fig. 5): ω m,i Ω m = ω m,i ω m,j, 3.8) j nb mi) where nb m i) := {j I m \ {i} : λ 2 ω m,i ω m,j ) 0} is the set of m-neighbors of i. Moreover, it is assumed that: DA-5) For each i I, j nbi) := {j I \ {i} : λ 2 ω i ω j ) 0}: x i x x j and j x i x i x j = n ωi ωi ω j, where 2 denotes Euclidian distance, 2 and n ωi ωi ω j is the restriction of the normal vector n ωi to the interface ω i ω j. Thus, the line segment joining neighboring vertices x i and x j is always perpendicular to ω i ω j see Fig. 5, where the vertices x i are chosen such that DA-5) is satisfied). 10

13 The approximation of the integrals over ω m,i Ω m, is now provided by replacing the normal gradient of θ ν on ω i ω j by the corresponding difference quotient θ ν n ωm,i θ ν,j θ ν,i ) λ 2 ωm,i ω m,j. 3.9) ω m,i Ω m x i x j 2 j nb m i) Approximation 3.9) is exact if θ ν is linear on the line segment connecting x i and x j. We now come to the discretization of the nonlocal terms. The approximation of the source term then follows in Sec. 3.4 below. 3.3 Discretization of Nonlocal Radiation Terms Similarly to the finite volume approximation of the local terms, the discretization of K Γ R Γ θ ν 1, θ ν )) and KRθ ν 1, θ ν )) proceeds by partitioning the surface of the respective radiation region i.e. Γ for K Γ R Γ θ ν 1, θ ν )) and Σ for KRθ ν 1, θ ν ))) into 2-dimensional polyhedral control volumes so-called boundary elements). DA-6) For a chosen fixed index ph, ζ α ) α IΩ and ζ α ) α IΣ are finite partitions see Def. 2.2) of Γ Ω and Σ, respectively, where I Ω I Σ =, ph / I Ω I Σ, 3.10) and, for each α I Ω resp. α I Σ ), the boundary element ζ α is a nonvoid, polyhedral, connected, and relative) open subset of Γ Ω resp. Σ), lying in a 2-dimensional affine subspace of R 3. For the convenience of subsequent concise notation, let ζ ph := Γ ph and I Γ := I Ω {ph}. On both Γ Ω and Σ, the boundary elements are supposed to be compatible with the control volumes ω i : DA-7) For each α I Ω resp. α I Σ ), there is a unique iα) I such that ζ α ω iα) Γ Ω resp. ζ α ω s,iα) Γ Σ ). Moreover, for each α I Ω I Σ : x iα) ζ α s. Fig. 6). Definition and Remark 3.2. For each i I, define J Ω,i := {α I Ω : λ 2 ζ α ω i ) 0} and J Σ,i := {α I Σ : λ 2 ζ α ω s,i ) 0}. It then follows from DA-1), DA-6), and DA-7), that ζ α ω i ) α JΩ,i is a partition of ω i Γ Ω = ω s,i Γ Ω and that ζ α ω s,i ) α JΣ,i is a partition of ω s,i Σ = ω i Σ s. Fig. 6). Moreover, A-2) implies that at most one of the two sets J Ω,i, J Σ,i can be nonvoid cf. Rem. 3.1 above). In the following, the discretization of K Γ R Γ θ ν 1, θ ν )) is considered. The procedure is analogous for KRθ ν 1, θ ν )), except slightly simpler, since it does not involve the phantom closure Γ ph. 11

14 x 1 ω 1 ω 2 ζ 1 ζ 2 x 2 i1) = 1, i2) = i3) = 2, i4) = 3, i5) = i6) = 4, i7) = 5 Γ ph ζ 7 ζ 3 ζ 4 ζ 5 ζ 6 x 4 ω 3 x 3 J Ω,1 = {1}, J Ω,2 = {2, 3}, J Ω,3 = {4}, J Ω,4 = {5, 6}, J Ω,5 = {7} x 5 ω 5 ω 4 Figure 6: Magnification of the open radiation region O 1 and of the adjacent part of Ω s cf. Figures 1, 2). It illustrates the partitioning of Γ Ω into the ζ α. In particular, it illustrates the compatibility condition DA-7) as well as Def. and Rem The radiosity R Γ θ ν 1, θ ν ) is approximated as constant on each boundary element ζ α, α I Ω. The approximated value is denoted by R α u ν 1, u ν ), depending on the vectors u ν 1 := θ ν 1,iβ) ) β IΩ, u ν := θ ν,iβ) ) β IΩ. On Γ ph, R Γ θ) = σ θ 4 ext by 2.18). Therefore, the K Γ -analogues of 2.7) and 2.8) yield ζ α K Γ R Γ θ ν 1, θ ν )) β I Ω R β u ν 1, u ν ) Λ α,β + σ θ 4 ext Λ α,ph α I Ω ), 3.11) where Λ α,β := Λ ω ζ α ζ β α, β) IΓ I Γ ). 3.12) Remark 3.3. Since points on the same boundary element ζ α can never see each other, Λ vanishes on ζ α ζ α, such that Λ α,α = 0. However, this fact will not be exploited in the following since we want to present the theory in a way that translates directly to the axisymmetric case, where, in general, Λ α,α > 0 cf. Sec. 3.6 below). The Λ α,β are nonnegative since Λω is nonnegative [Tii97b, Lem. 2]). The forms of Λ and ω imply the symmetry condition Λ α,β = Λ β,α α, β) IΓ I Γ ). 3.13) Since Γ = Γ Ω Γ ph is a closed surface, the conservation of radiation energy 2.10) yields β I Γ Λ α,β = λ 2 ζ α ) α I Ω ). 3.14) Using 3.11) allows to write 2.11) in the integrated and discretized form R α u ν 1, u ν ) λ 2 ζ α ) 1 ɛθ ν 1,iα) ) ) β I Ω R β u ν 1, u ν ) Λ α,β = σ ɛθ ν 1,iα) ) θ 4 ν,iα) λ 2 ζ α ) + σ 1 ɛθ ν 1,iα) ) ) θ 4 ext Λ α,ph α I Ω ) )

15 If the vectors u ν 1 = θ ν 1,iα) ) α IΩ and u ν = θ ν,iα) ) α IΩ are known, then 3.15) constitutes a linear system for the determination of the vector R α u ν 1, u ν )) α IΩ. In matrix form, 3.15) reads with vector-valued functions Gu ν 1 ) Ru ν 1, u ν ) = Eu ν 1, u ν ) + E ph u ν 1 ), 3.16) R : R + 0 ) I Ω R + 0 ) I Ω R + 0 ) I Ω, Rũ, u) = R α ũ, u) ) α I Ω, 3.17a) E : R + 0 ) I Ω R + 0 ) I Ω R + 0 ) I Ω, Eũ, u) = E α ũ, u) ) α I Ω, E α ũ, u) :=σ ɛũ α ) u 4 α λ 2 ζ α ), E ph : R + 0 ) I Ω R + 0 ) I Ω, E ph ũ) = E ph,α ũ) ) α I Ω, E ph,α ũ) :=σ 1 ɛũ α ) ) θ 4 ext Λ α,ph, 3.17b) 3.17c) R is indeed nonnegative, see 3.18) and the proof of Lem. 3.7a) below), and a matrix-valued function G : R + 0 ) I Ω R I2 Ω, Gũ) = Gα,β ũ) ) α,β) I { Ω, 2 λ 2 ζ α ) 1 ɛũ α ) ) Λ α,β for α = β, G α,β ũ) := 1 ɛũ α ) ) Λ α,β for α β. 3.17d) Lemma 3.4. The following holds for each u R + 0 ) I Ω : a) For each α I Ω : β I Ω \{α} G α,βu) 1 ɛu α )) G α,α u) < G α,α u). In particular, Gu) is strictly diagonally dominant. b) Gu) is an M-matrix, i.e. Gu) is invertible, G 1 u) is nonnegative, and G α,β u) 0 for each α, β) I 2 Ω, α β. Proof. a): Combining 3.17d) with 3.14) yields G α,β u) 1 ɛuα ) ) Λ α,β β I Ω \{α} β I Γ \{α} = 1 ɛu α ) ) λ 2 ζ α ) Λ α,α ) α I Ω ), proving a) since ɛ > 0. b): According to 3.17d), the nonnegativity of the Λ α,β yields that G α,β u) 0 for α β, whereas a) shows that G α,α u) > 0. Since Gu) is also strictly diagonally dominant according to a), Gu) is an M-matrix by [Axe94, Lem. 6.2]. Remark 3.5. If one were to relax A-6) to allow ɛθ) = 0, thereby admitting completely reflecting and not emitting parts of the surface, then one could no longer expect Gu) to be strictly diagonally dominant. However, as long as there is no 13

16 connected radiation region where ɛ vanishes identically, Gu) is still weakly diagonally dominant, and one can still prove Lem. 3.4b) using [Col68, 23, Th. 2]. In consequence, the subsequent development can still be carried out and Lem. 3.7 can still be proved. If ɛ did vanish identically within some connected radiation region, then, on the region s surface, one had to remove R and J from the corresponding interface condition. Now, Lemma 3.4b) allows to give a precise definition of R by completing 3.17a) with Rũ, u) := G 1 ũ) Eũ, u) + E ph ũ) ). 3.18) Remark 3.6. The definition of R in 3.18) implies that 3.15) and 3.16) hold with u ν 1 = θ ν 1,iα) ) α IΩ and u ν = θ ν,iα) ) α IΩ replaced by general vectors ũ = ũ α ) α IΩ R + 0 ) I Ω and u = u α ) α IΩ R + 0 ) I Ω, respectively. Finally, introducing the vector-valued function V Γ : R + 0 ) I Ω R + 0 ) I Ω R + 0 ) I Ω, V Γ ũ, u) = V Γ,α ũ, u) ) α I Ω, V Γ,α ũ, u) := ɛũ α ) β I Ω R β ũ, u) Λ α,β + σ ɛũ α ) θ 4 ext Λ α,ph, 3.19) 3.11) provides the desired approximation of the nonlocal term in 3.7a): ɛθ ν 1 ) K Γ R Γ θ ν 1, θ ν )) ζ α ɛθ ν 1,iα) ) β I Ω R β u ν 1, u ν ) Λ α,β + σ ɛθ ν 1,iα) ) θ 4 ext Λ α,ph = V Γ,α u ν 1, u ν ). 3.20) Working with the partition ζ α ) α IΣ of Σ, a procedure analogous to the one described above where Σ plays the role of Γ Ω, and Γ ph = ) leads to the definition of a vectorvalued function V Σ : R + 0 ) I Σ R + 0 ) I Σ R + 0 ) I Σ, V Σ ũ, u) = V Σ,α ũ, u) ), α I Σ providing the approximation of the nonlocal term in 3.7b): ɛθ ν 1 ) KRθ ν 1, θ ν )) ɛθ ν 1,iα) ) R β u ν 1, u ν ) Λ α,β = V Σ,α u ν 1, u ν ). ζ α β I Σ 3.21) For subsequent use, the following Lem. 3.7 states some properties of the functions V Γ and V Σ. We introduce the following notation for u = u i ) i I R I where I can be an arbitrary, nonempty, finite index set): min u) := min{u i : i I}, max u) := max{u i : i I}. 3.22) Lemma 3.7. a) Both V Γ and V Σ are nonnegative. 14

17 b) For each ũ, u) R + 0 ) I Ω R + 0 ) I Ω, α I Ω : σ ɛũ α ) min { min u) 4, θ 4 ext} λ2 ζ α ) V Γ,α ũ, u) and, for each ũ, u) R + 0 ) I Σ R + 0 ) I Σ, α I Σ : σ ɛũ α ) max { max u) 4, θ 4 ext} λ2 ζ α ), σ ɛũ α ) min u) 4 V Σ,α ũ, u) σ ɛũ α ) max u) 4 λ 2 ζ α ). c) For each r R + and ũ R + 0 ) I Ω, with respect to the max-norm, the map V Γ,α ũ, ) is 4 σ ɛũ α ) λ 2 ζ α ) Λ α,ph ) r 3) -Lipschitz on [0, r] I Ω. Analogously, for each r R + and ũ R + 0 ) I Σ, with respect to the max-norm, the map V Σ,α ũ, ) is 4 σ ɛũ α ) λ 2 ζ α ) r 3) -Lipschitz on [0, r] I Σ. Proof. a): Since 0 < ɛ 1, E and E ph are nonnegative by 3.17b) and 3.17c), respectively. Then R is nonnegative according to 3.18) and Lem. 3.4b). The nonnegativity of V Γ is now a direct consequence of 3.19). An analogous argument shows V Σ 0. b): Note that, since Rũ, u) satisfies 3.15) by Rem. 3.6, one has R α ũ, u) λ 2 ζ α ) 1 ɛũ α ) ) R β ũ, u) Λ α,β β I Ω σ max { max u) 4 }, θext 4 ɛũ α ) λ 2 ζ α ) + 1 ɛũ α ) ) ) Λ α,ph = σ max { max u) 4 }, θext 4 λ 2 ζ α ) 1 ɛũ α ) ) ) i.e. Gũ) Rũ, u) Gũ) U max, where β I Ω Λ α,β U max = U max,α ) α IΩ, U max,α := σ max { max u) 4, θ 4 ext}, α I Ω ), 3.23) implying Rũ, u) U max, as G 1 ũ) 0 by Lem. 3.4b). Thus, R α ũ, u) σ max { max u) 4, θext} 4 for each α IΩ. Likewise, one obtains that R α ũ, u) σ min { min u) 4, θext} 4 for each α IΩ. The estimates for V Γ,α ũ, u) now follow from 3.19) by combining the estimates for R α ũ, u) with 3.14). An analogous argument shows the second part of b). c): Observe that the function θ λ θ 4 is 4λr 3 )-Lipschitz on [0, r], such that, by 3.15), for each ũ, u, v) [0, r] I Ω [0, r] I Ω [0, r] I Ω, α I Ω : Rα ũ, u) R α ũ, v) ) λ 2 ζ α ) 1 ɛũ α ) ) Rβ ũ, u) R β ũ, v) ) Λ α,β β I Ω = σ ɛũ α ) u 4 α vα 4 λ2 ζ α ) 4 σ ɛũ α ) u α v α λ 2 ζ α ) r ) 15

18 Now, let α I Ω be such that N max := Rũ, u) Rũ, v) max = R α ũ, u) R α ũ, v). Then 3.24) implies 4 σ ɛũ α ) u v max λ 2 ζ α ) r ) N max λ 2 ζ α ) 1 ɛũ α ) ) Rβ ũ, u) R β ũ, v) ) Λ α,β β I Ω Lem. 3.4a) N max λ 2 ζ α ) 1 ɛũ α ) ) Rβ ũ, u) R β ũ, v) ) Λ α,β β I Ω N max λ 2 ζ α ) 1 ɛũ α ) ) ) 3.14) N max ɛũ α ) λ 2 ζ α ), β I Ω Λ α,β showing that Rũ, ) is 4 σ r 3 )-Lipschitz on [0, r] I Ω. The claimed Lipschitz continuity of V Γ ũ, ) now follows from 3.19). An analogous argument shows the second part of c). 3.4 Approximation of the Source Term and of the Initial Condition For the approximation of the source term, let tν t f m,ν,i ν 1 ω m,i f m k ν λ 3 ω m,i ) 3.25) be a suitable approximation, where, in general, the choice will depend on the regularity of f m for f m continuous, one might choose f m,ν,i := f m t ν, x i ), but f m,ν,i := k ν λ 3 ω m,i )) 1 t ν t ν 1 ω m,i f m for a general f m L 0, T, L Ω m ))). However, a suitable approximation is assumed to satisfy: AA-1) For each m {s, g}, ν {0,..., N}, and i I: 0 f m,ν,i f m L t ν 1, t ν, L ω m,i )). Remark 3.8. If A-5) holds, then f m,ν,i := k ν λ 3 ω m,i )) 1 t ν t ν 1 ω m,i f m guarantees AA-1). If f m is continuous, then AA-1) is also satisfied for f m,ν,i := f m t ν, x i ). Let θ init,i be a suitable approximation of θ init on ω i, i I. For a continuous θ init, one might choose θ init,i := θ init x i ), in contrast to θ init,i := λ 3 ω i )) 1 ω i θ init for a general θ init L Ω, R + 0 ). A suitable approximation is assumed to satisfy: AA-2) For each i I: 0 ess infθ init ωi ) θ init,i θ init L ω i,r + 0 ), where ess infθ init ωi ) denotes the essential infimum of θ init on the set ω i. Remark 3.9. AA-2) is satisfied for θ init,i = θ init x i ) for a continuous θ init ) and for θ init,i = λ 3 ω i )) 1 ω i θ init for a general θ init ). 16

19 3.5 The Finite Volume Scheme For u = u i ) i I, define u IΩ := u iα) ) α IΩ, u IΣ := u iα) ) α IΣ. 3.26) At this point, all preparations are in place to state the finite volume scheme in 3.27) and 3.28) below. The terms in 3.28) arise from 3.2) after summing over m {s, g} and employing the approximations 3.4), 3.6), 3.9), 3.25), 3.20), and 3.21), respectively. One is seeking a nonnegative solution u 0,..., u N ), u ν = u ν,i ) i I, to u 0,i = θ init,i i I), 3.27a) H ν,i u ν 1, u ν ) = 0 i I, ν {1,..., N}), 3.27b) where, for each ν {1,..., N}: H ν,i : R + 0 ) I R + 0 ) I R, H ν,i ũ, u) = kν 1 εm u i ) ε m ũ i ) ) λ 3 ω m,i ) 3.28a) κ m j nb m i) + σ ɛũ i ) u 4 i λ 2 ωs,i Γ Ω ) u j u i ) λ 2 ωm,i ω m,j x i x j 2 α J Ω,i V Γ,α ũ IΩ, u IΩ ) 3.28b) 3.28c) + σ ɛũ i ) u 4 i θ 4 ext) λ 2 ωs,i Ω \ Γ Ω ) ) 3.28d) + σ ɛũ i ) u 4 i λ 2 ωi Σ ) f m,ν,i λ 3 ω m,i ). α J Σ,i V Σ,α ũ IΣ, u IΣ ) In general, many summands in 3.28) vanish, e.g. if ω i Ω g and ω s,i =. 3.28e) 3.28f) 3.6 Modifications for the Axisymmetric Case Suppose the space domains Ω s and Ω g are axisymmetric, and, in cylindrical coordinates r, ϑ, z), the considered space-dependent functions here: θ, f s and f g ) are independent of the angular coordinate ϑ. Then the circular projection r, ϑ, z) r, z) can be used to reduce the model of Sec. 2 as well as the finite volume scheme to two space dimensions. For the nonlocal radiation terms R and J see Sections 2.2, 2.3, and 3.3 above), the dimension reduction for the axisymmetric case was carried out in [Phi03, Sections 2.4.3, 3.7.8]. Even though the cylindrical symmetry affects the calculation of visibility and view factors, the essential properties of the radiation matrices proved in Lemmas 3.4 and 17

20 3.7 persist. We stress once more that, in our reasoning above, we have not used Λ α,α = 0, as, in general, it is not valid in the axisymmetric case. In a more general context, it was shown in [Phi03, Sec. 3.6], how symmetry conditions together with a change of variables can be used to reduce the space dimension in a finite volume scheme. In the case of cylindrical coordinates, the change of variables merely yields a factor r in the integrands occurring in 3.4), 3.6), 3.9), 3.20), and 3.21), and thus in the corresponding terms in 3.28). In consequence, for the axisymmetric finite volume scheme, analogous reasoning to the contents of the following Section 4 can still be used to prove a maximum principle as well as existence and uniqueness for the discrete solution, analogous to Th. 4.3, Cor. 4.4, Th. 4.5, and Rem. 4.6 below. 4 Discrete Existence and Uniqueness 4.1 A Root Problem with Maximum Principle The proof of the existence and uniqueness of a discrete solution to the finite volume scheme 3.27) in Th. 4.3 and Th. 4.5 below is based on the solution to the root problem in Th. 4.2 below. Theorem 4.2 establishes a maximum principle for roots to a certain type of continuous discrete nonlinear operator H. The maximum principle is a consequence of the assumption that the components H i of H can be decomposed into scalar-dependent continuous functions b i and h i, and a vector-dependent continuous function g i such that the b i are sufficiently increasing, and g i h i satisfy the boundedness condition in Th. 4.2ii). Existence and uniqueness of the solution to the root problem in Th. 4.2 is founded on the following Lem. 4.1, providing a unique root to continuous functions H : [m, M] I R I, presuming the components H i of H can be decomposed into the difference of a scalar-dependent, sufficiently increasing function h i and a vectordependent, Lipschitz continuous function g i. Lemma 4.1. Let m, M R with m < M. Given a finite, nonempty index set I, consider an operator H : [m, M] I R I, Hu) = H i u) ) i I. 4.1) Assume there are continuous functions h i C[m, M], R), g i C[m, M] I, R), i I, and families of numbers L g,i ) i I R + 0 ) I, C h,i ) i I R + ) I, such that the following conditions i) v) are satisfied. i) For each i I, u [m, M] I : H i u) = h i u i ) g i u). ii) For each i I, u [m, M] I : h i m) g i u) h i M). iii) Each g i, i I, is L g,i -Lipschitz with respect to the max-norm on [m, M] I. 18

21 iv) For each i I and M θ 2 θ 1 m: h i θ 2 ) θ 2 θ 1 ) C h,i + h i θ 1 ). v) L g,i < C h,i for each i I. Then H has a unique root in [m, M] I, i.e. there is a unique u 0 [m, M] I such that Hu 0 ) = 0, where 0 := 0,..., 0). Proof. Define f : [m, M] I [m, M] I, f i := h 1 i g i. 4.2) It is noted that the h 1 i exist on [h i m), h i M)], as the h i are assumed continuous, as well as strictly increasing on [m, M] by iv). Moreover, h 1 i can be composed with g i by ii). According to iv), h 1 i is C 1 h,i -Lipschitz, which, together with iii) and v), implies that each f i is L g,i C h,i -contracting. Then f is also contracting and the Banach fixed point theorem yields that f has a unique fixed point u 0 = u 0,k ) i I [m, M] I. According to i), iv), and 4.2), u 0 is a fixed point of f if, and only if, u 0 is a root of H, i.e. the proof is complete. Theorem 4.2. Let τ R be a closed, open, half-open, bounded or unbounded) interval. Given a finite, nonempty index set I, and given ũ τ I, consider a continuous operator H : τ I R I, Hu) = H i u) ) i I. 4.3) Assume there are continuous functions b i Cτ, R), h i Cτ, R), g i Cτ I, R), i I, such that the following conditions i) iii) are satisfied. i) There is ũ τ I such that, for each i I, u τ I : H i u) = b i u i ) + h i u i ) b i ũ i ) g i u). ii) There are m, M τ, a family of nonpositive numbers β i ) i I R 0 ) I, and a family of nonnegative numbers B i ) i I R + 0 ) I such that, for each i I, u τ I, θ τ: max { max u), M} θ gi u) h i θ) B i, 4.4a) θ min { m, min u) } g i u) h i θ) β i, 4.4b) where max u) and min u) are according to 3.22). iii) There is a family of positive numbers C b,i ) i I R + ) I such that, for each i I and θ 1, θ 2 τ: θ 2 θ 1 b i θ 2 ) θ 2 θ 1 ) C b,i + b i θ 1 ). Letting { } { } βi Bi β := min : i I, B := max : i I, 4.5) C b,i C b,i mũ) := min { m, min ũ) + β }, Mũ) := max { M, max ũ) + B }, 4.6) 19

22 we have the following maximum pinciple: If u 0 τ I satisfies Hu 0 ) = 0 := 0,..., 0), then u 0 [mũ), Mũ)] I. If, in addition to i) iii), the following conditions iv) vi) are satisfied, then there is a unique u 0 [mũ), Mũ)] I such that Hu 0 ) = 0. iv) For each i I, there is L g,i ũ) R + 0 such that g i is L g,i ũ)-lipschitz with respect to the max-norm on [mũ), Mũ)] I. v) For each i I, there is C h,i ũ) R + 0 such that, for each θ 1, θ 2 [mũ), Mũ)]: θ 2 θ 1 h i θ 2 ) θ 2 θ 1 ) C h,i ũ) + h i θ 1 ). vi) L g,i ũ) < C b,i + C h,i ũ) for each i I. Proof. We start by showing that, given i) iii), each root of H must lie in [mũ), Mũ)] I. Consider u τ I, max u) > Mũ). Let i I be such that u i = max u). Then, since u i > Mũ) M, 4.4a) applies with θ = u i, yielding g i u) h i u i ) B i. 4.7) Moreover, since u i > Mũ) max ũ) + B ũ i, one can apply iii) with θ 2 = u i and θ 1 = ũ i to get b i u i ) u i ũ i ) C b,i + b i ũ i ). 4.8) Combining 4.7) and 4.8) with i), we find H i u) u i ũ i ) C b,i B i > ũ i + B ũ i ) C b,i B i 0, 4.9) i.e. u is not a root of H. An analogous argument shows that, if u τ I and min u) < mũ), then u is not a root of H, concluding the proof that each root of H must lie in [mũ), Mũ)] I. It remains to show that H has a unique root in [mũ), Mũ)] I. This is done by an application of Lem If, for i I, h i C [mũ), Mũ)], R ), h i := b i + h i, g i C [mũ), Mũ)] I, R ), g i u) := b i ũ i ) + g i u), then i) immediately implies condition i) of Lem The verification of conditions ii) v) of Lem. 4.1 is the remaining task of this proof. Lem. 4.1ii): One has to show b i mũ) ) + hi mũ) ) bi ũ i ) + g i u) b i Mũ) ) + hi Mũ) ) 4.10) for each u [mũ), Mũ)] I, i I. Since mũ) m, one can apply 4.4b) with θ = mũ), and, since mũ) ũ i, one can apply iii) with θ 1 = mũ) and θ 2 = ũ i. This yields b i mũ) ) + hi mũ) ) bi ũ i ) ũ i mũ) ) C b,i +g i u) β i b i ũ i )+ g i u), 4.11) where the last inequality is due to mũ) ũ i + β i C b,i. The first inequality of 4.10) is proved by 4.11), and an analogous argument shows the second inequality of 4.10). 20

23 Lem. 4.1iii): Each g i, i I, is L g,i := L g,i ũ)-lipschitz with respect to the maxnorm on [mũ), Mũ)] I, since g i is L g,i -Lipschitz with respect to the max-norm on [mũ), Mũ)] I according to hypothesis iv). Lem. 4.1iv): Letting C h,i := C b,i + C h,i ũ), for mũ) θ 1 θ 2 Mũ), one has to verify h i θ 2 ) θ 2 θ 1 ) C h,i + h i θ 1 ). 4.12) Since h i = b i + h i on [mũ), Mũ)], 4.12) follows by adding the conditions in iii) and v). Lem. 4.1v): By hypothesis vi), one has L g,i = L g,i ũ) < C b,i + C h,i ũ) = C h,i for each i I as needed. Since all hypotheses of Lem. 4.1 are verified, Lem. 4.1 grants that H has a unique root in [mũ), Mũ)] I, thereby concluding the proof of Th Existence and Uniqueness of a Discrete Solution to the Finite Volume Scheme, Maximum Principle The following Th. 4.3 is the main building block for all the discrete existence and uniqueness results provided subsequently. Theorem 4.3 can be considered as a discrete existence result with maximum principle, locally in time. Given an arbitrary vector ũ R + 0 ) I, Th. 4.3 establishes that each root of the finite volume scheme operator H ν ũ, ) of 3.28) satisfies a maximum principle. Moreover, Th. 4.3 proves the existence of a unique root to H ν ũ, ), provided that the ν-th time step k ν is sufficiently small. The upper and lower bound for the solution, respectively given by 4.13c) and 4.13d) below, are determined by the external temperature θ ext, by the max and min of ũ as defined in 3.22), by the size of the time step, and by the values of the heat sources in the time interval [t ν 1, t ν ]. The condition on the time step size 4.15) arises from the nonlocal terms in 3.28), namely, 3.28b), 3.28c), and 3.28d). It depends on the constant L V defined in 4.13b) below, involving the ratios between the size of boundary elements and adjacent volume elements. Thus, L V is of order h 1 if h is a parameter for the fineness of a space discretization constructed by uniform refinement of some initial grid. Letting ũ = u ν 1, as a direct consequence of Th. 4.3, for k ν small enough, each nonnegative solution u 0,..., u ν 1 ) to the finite volume scheme 3.27) with N replaced by ν 1 < N, can be uniquely extended to t = t ν s. Cor. 4.4). Finally, in Th. 4.5, we use an inductive argument to show that condition 4.15) and the bounds from the maximum principle are sufficiently benign to guarantee a unique solution to the entire finite volume scheme 3.27). Theorem 4.3. Assume A-1) A-8), DA-1) DA-7), AA-1) and AA-2). 21

24 Moreover, assume ν {1,..., N} and ũ = ũ i ) i I ) I. R + 0 Let B f,ν := max f m,ν,i λ3ω m,i ) : i I λ 3 ω i ), λ 2 ω i Σ) L V := 4 σ max + λ 2 ζ α ) Λ α,ph : i I λ 3 ω i ) λ 3 ω i ), α J Ω,i 4.13a) 4.13b) mũ) := min { θ ext, min ũ) }, 4.13c) { M ν ũ) := max θ ext, max ũ) + k } ν B f,ν, 4.13d) C ε with min ũ), max ũ) according to 3.22), and C ε according to A-3). Then we have the maximum principle that each solution u ν = u ν,i ) i I ) R + I 0 to H ν,i ũ, u ν ) = 0 i I) 4.14) must lie in [mũ), M ν ũ)] I. Furthermore, if k ν is such that k ν Mν ũ) 3 mũ) 3) L V < C ε, 4.15) then there is a unique u ν [mũ), M ν ũ)] I satisfying 4.14). Proof. Before starting with the main part of the proof, we would like to point out that, by choosing k ν sufficiently small, one can ensure that 4.15) is satisfied. Now, the goal is to apply Th. 4.2 with τ = R + 0 and H ν ũ, ) playing the role of H. To that end, we will define continuous functions b ν,i, h i, g ν,i, as well as numbers m, M R + 0, β i R 0, B ν,i R + 0, C b,ν,i R +, L g,ν,i ũ) R +, and C h,ν,i ũ) R + that satisfy the hypotheses of Th. 4.2 where the quantities with index ν correspond to the matching quantities without index ν in Th. 4.2). Condition 4.15) will only be needed to prove hypothesis vi) of Th For each i I, let b ν,i : R + 0 R + 0, b ν,i θ) := k 1 ν L κ,i := κ m j nb m i) C V,i ũ) := σ ɛũ i ) λ 2 ωs,i Γ Ω ) λ 2 ωm,i ω m,j ) ε m θ) λ 3 ω m,i ), 4.16a) x i x j 2 0, 4.16b) + σ ɛũ i ) λ 2 ωs,i Ω \ Γ Ω ) ) + σ ɛũ i ) λ 2 ω i Σ) 0, 4.16c) h i : R + 0 R + 0, hi θ) := θ L κ,i + θ 4 C V,i ũ), 4.16d) 22

25 g ν,i : R + 0 ) I R + 0, g ν,i u) := + κ m j nb m i) u j x i x j 2 λ 2 ω m,i ω m,j ) V Γ,α ũ IΩ, u IΩ ) + σ ɛũ i ) θext 4 λ 2 ωs,i Ω \ Γ Ω ) ) α J Ω,i + V Σ,α ũ IΣ, u IΣ ) + f m,ν,i λ 3 ω m,i ), α J Σ,i m := M := θ ext, β i := 0, B ν,i := f m,ν,i λ 3 ω m,i ), 4.16e) 4.16f) C b,ν,i := kν 1 C ε λ 3 ω i ) > 0, 4.16g) L V,i ũ) := 4 σ ɛũ i ) λ 2 ζ α ) Λ α,ph ) + λ 2 ω i Σ) 0, 4.16h) α J Ω,i L g,ν,i ũ) := M ν ũ) 3 L V,i ũ) + L κ,i 0, C h,ν,i ũ) := C b,ν,i + L κ,i + 4 mũ) 3 C V,i ũ) > i) 4.16j) Claim 1. For each i I, ũ R + 0 ) I, the numbers L κ,i, C V,i ũ), L V,i ũ), L g,ν,i, and the functions h i, g ν,i are indeed nonnegative; the numbers C b,ν,i and C h,ν,i ũ) are indeed positive. Proof. The assumed nonnegativity of κ m, σ, and ɛ implies that all summands in 4.16b) and 4.16c) are nonnegative, proving the nonnegativity of L κ,i, C V,i ũ), and h i. Throwing in 3.14), the nonnegativity of L V,i ũ) and L g,ν,i is immediate from their definitions in 4.16h) and 4.16i), respectively. Using Lem. 3.7a), A-5) A- 7), and AA-1), one sees that g ν,i 0. Finally, since C ε, k ν, and λ 3 ω i ) are positive, so are C b,ν,i and C h,ν,i ũ) by 4.16g) and 4.16j), respectively. Claim 2. The numbers mũ) and M ν ũ) defined in 4.13c) and 4.13d), respectively, correspond to the numbers mũ) and Mũ) as defined in 4.6) in Th Proof. Since, for each i I, β i = 0 according to 4.16f), one has β = 0 by 4.5), showing mũ) = min { θ ext, min ũ) + β }. Since, for each i I, B ν,i := f m,ν,i λ 3 ω m,i ) according to 4.16f), one has { } Bν,i B = B ν := max : i I = k ν B f,ν C b,ν,i C ε { by 4.5), 4.13a), and 4.16g), showing M ν ũ) = max θ ext, max ũ) + kν C ε B f,ν }. The hypotheses i) vi) of Th. 4.2 are now verified consecutively. 23

26 Th. 4.2i): To show H ν,i ũ, u) = b ν,i u i ) + h i u i ) b ν,i ũ i ) g ν,i u), observe b ν,i u i ) b ν,i ũ i ) = k 1 ν εm u i ) ε m ũ i ) ) λ 3 ω m,i ), and definitions 4.16d), 4.16b), 4.16c), and 4.16e) are designed such that h i u i ) g ν,i u) = H ν,i ũ, u) kν 1 εm u i ) ε m ũ i ) ) λ 3 ω m,i ). Th. 4.2ii): One has to show that, for each i I, u R + 0 ) I, θ R + 0 : max { max u), θ ext } θ gν,i u) h i θ) B ν,i, 4.17a) θ min { θ ext, min u) } g ν,i u) h i θ) b) Considering Lem. 3.7b) and Def. and Rem. 3.2, we see that V Γ,α ũ IΩ, u IΩ ) σ ɛũ i ) max { max u) 4 }, θext 4 λ2 ω s,i Γ Ω ), α J Ω,i V Σ,α ũ IΣ, u IΣ ) σ ɛũ i ) max u) 4 λ 2 ω i Σ). α J Σ,i If θ θ ext and θ max u), then, by recalling 4.13a) and 4.16b) 4.16f), we have g ν,i u) κ m j nb mi) θ ) λ 2 ωm,i ω m,j x i x j 2 + σ ɛũ i ) θ 4 λ 2 ω s,i Γ Ω ) + σ ɛũ i ) θ 4 λ 2 ωs,i Ω \ Γ Ω ) ) + σ ɛũ i ) θ 4 λ 2 ω i Σ) + = θ L κ,i + θ 4 C V,i ũ) + f m,ν,i λ 3 ω m,i ) f m,ν,i λ 3 ω m,i ) = h i θ) + B ν,i, proving 4.17a). On the other hand, if θ θ ext and θ min u), then, as f m,ν,i 0 by AA-1), an analogous computation shows g ν,i u) θ L κ,i + θ 4 C V,i ũ) = h i θ), proving 4.17b). Th. 4.2iii): That, for θ 2 θ 1 0, one has b ν,i θ 2 ) θ 2 θ 1 ) C b,ν,i + b ν,i θ 1 ) is immediate from combining A-3), 4.16a), and 4.16g). Th. 4.2iv): For each i I, one has to show that g ν,i is L g,ν,i ũ)-lipschitz with respect to the max-norm on [mũ), M ν ũ)] I. The function u κ m j nb m i) u j x i x j 2 λ 2 ω m,i ω m,j ) u R + 0 ) I) 24

27 is L κ,i -Lipschitz, L κ,i according to 4.16b). Lemma 3.7c) and Def. and Rem. 3.2 show that α J Ω,i V Γ,α ũ IΩ, ) is 4σ ɛũ i ) M ν ũ) 3 λ 2 ω s,i Γ Ω ) α J Ω,i Λ α,ph )- Lipschitz on [0, M ν ũ)] I Ω and that α J Σ,i V Σ,α ũ IΣ, ) is 4σ ɛũ i ) M ν ũ) 3 λ 2 ω i Σ)- Lipschitz on [0, M ν ũ)] I Σ. Recalling 4.16h) yields that the function u V Γ,α ũ IΩ, u IΩ ) + V Σ,α ũ IΣ, u IΩ ) α J Ω,i α J Σ,i is M ν ũ) 3 L V,i -Lipschitz on [0, M ν ũ)] I. Therefore, by 4.16e) and 4.16i), g ν,i is L g,ν,i ũ)-lipschitz on [mũ), M ν ũ)] I as needed. Th. 4.2v): Let i I and M ν ũ) θ 2 θ 1 mũ). We need to show that h i θ 2 ) θ 2 θ 1 ) L κ,i + 4 mũ) 3 C V,i ũ) ) + h i θ 1 ). Since θ θ 4 is a convex function on R + 0, one has θ mũ) 3 θ 2 + θ 1 ) + θ 4 1. As C V,i ũ) 0, recalling 4.16d) yields h i θ 2 ) θ 2 θ 1 )L κ,i + θ 1 L κ,i + 4 mũ) 3 θ 2 θ 1 ) C V,i ũ) + θ 4 1 C V,i ũ) = θ 2 θ 1 ) L κ,i + 4 mũ) 3 C V,i ũ) ) + h i θ 1 ), thereby establishing the case. Th. 4.2vi): For each i I, one has to show that L g,ν,i ũ) < C h,ν,i ũ), where L g,ν,i ũ) and C h,ν,i ũ) are according to 4.16i) and 4.16j), respectively. Taking into account 4.13b), 4.16h), and A-6), we have L V,i ũ) L V λ 3 ω i ). Moreover, recalling Λ α,ph 0 for each α J Ω,i, 4.16h), and Def. and Rem. 3.2, we obtain L V,i ũ) 4 σ ɛũ i ) λ 2 ω s,i Γ Ω ) + λ 2 ω i Σ) ) 4 C V,i ũ). These estimates for L V,i ũ), combined with 4.16i) and hypothesis 4.15), yield L g,ν,i ũ) M ν ũ) 3 mũ) 3) L V λ 3 ω i ) + mũ) 3 4 C V,i ũ) + L κ,i < C ε k ν λ 3 ω i ) + L κ,i + 4 mũ) 3 C V,i ũ), i.e., by 4.16g) and 4.16j), L g,ν,i ũ) < C h,ν,i ũ) as needed. Hence, all hypotheses of Th. 4.2 are verified, and the conclusion of Th. 4.2 provides a unique vector u ν [mũ), M ν ũ)] I such that H ν,i ũ, u ν ) = 0 for each i I. Since Th. 4.2 also yields that u ν is the only element of R + 0 ) I satisfying H ν,i ũ, u ν ) = 0 for each i I, the proof of Th. 4.3 is complete. Corollary 4.4. Assume A-1) A-8), DA-1) DA-7), AA-1), AA-2), and let u 0,..., u n 1 ), n N, u ν = u ν,i ) i I, be a nonnegative solution to 3.27) where N is replaced by n 1). Then each solution u n ) R + I 0 to Hn,i u n 1, u n ) = 0 for each i I), where H n,i is defined by 3.28), must lie in [mu n 1 ), M n u n 1 )] I, with mu n 1 ) and M n u n 1 ) defined according to 4.13c) and 4.13d), respectively. Furthermore, if k n satisfies condition 4.15), then there is a unique u n ) R + I 0 that satisfies H n,i u n 1, u n ) = 0 for each i I. 25

28 Theorem 4.5. Assume A-1) A-8), DA-1) DA-7), AA-1) and AA-2). Let M ν := max {θ ext, θ init Ω,R } L +0 ) + t ν C ε for each ν {0,..., N}. m := min { θ ext, ess infθ init ) }, 4.18) f m L 0,t ν,l Ω m )) 4.19) If u 0,..., u N ) = u ν,i ) ν,i) {0,...,N} I R + 0 ) I {0,...,N} is a solution to the finite volume scheme 3.27), then u ν [m, M ν ] I for each ν {0,..., N}. Furthermore, if k ν M 3 ν m 3) L V < C ε ν {1,..., N} ), 4.20) where L V is defined according to 4.13b), then the finite volume scheme 3.27) has a unique solution u 0,..., u N ) R + 0 ) I {0,...,N}. It is pointed out that a sufficient condition for 4.20) to be satisfied is max { k ν : ν {1,..., N} } M 3 N m 3) L V < C ε. 4.21) Proof. The proof is carried out by induction on n {0,..., N}. For n = 0, u 0,i = θ init,i for i I is uniquely determined by 3.27a). By AA-2), for each i I, one has m ess infθ init ) θ init,i θ init L ω i,r + 0 ) M 0, showing u 0 [m, M 0 ] I. Now, let N n > 0. Consider u 0,..., u n ) R + 0 ) I {0,...,n} satisfying 3.27) with N replaced by n. Then, by induction, we know u 0,..., u n 1 ) ν {0,...,n 1} [m, M ν] I, and it remains to show u n [m, M n ]. By AA-1): f m,n,i f m L t n 1, t n, L ω m,i )) f m L 0,t n,l Ω m )). 4.22) Using 4.22) in 4.13a), we infer B f,n = max f m,n,i λ3ω m,i ) λ 3 ω i ) : i I f m L 0,t n,l Ω m )). 4.23) Applying the induction hypothesis, and combining 4.23) with 4.13c) and 4.13d), yields m min { θ ext, min u n 1 ) } = mu n 1 ), { M n u n 1 ) = max θ ext, max u n 1 ) + k } n B f,n C ε M n 1 + k n f m L C 0,t n,l Ω m)) M n. ε 4.24a) 4.24b) Thus, if u n ) R + I 0 satisfies the equation Hn u n 1, u n ) = 0, then, according to Cor. 4.4 and 4.24), for each i I: m mu n 1 ) u n,i M n u n 1 ) M n, showing u n [m, M n ] I. 26