Evolving surface finite element methods for random advection-diffusion equations
Ana Djurdjevac, Charles M. Elliott, Ralf Kornhuber, Thomas Ranner
EEVOLVING SURFACE FINITE ELEMENT METHODS FORRANDOM ADVECTION-DIFFUSION EQUATIONS
ANA DJURDJEVAC, CHARLES M. ELLIOTT, RALF KORNHUBER,AND THOMAS RANNER
Abstract.
In this paper, we introduce and analyse a surface finite elementdiscretization of advection-diffusion equations with uncertain coefficients onevolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove optimal error bounds for the semi-discrete solutionand Monte-Carlo samplings of its expectation in appropriate Bochner spaces.Our theoretical findings are illustrated by numerical experiments in two andthree space dimensions. Introduction
Surface partial differential equations, i.e., partial differential equations on sta-tionary or evolving surfaces, have become a flourishing mathematical field withnumerous applications, e.g., in image processing [26], computer graphics [6], cellbiology [21, 34], and porous media [32]. The numerical analysis of surface partialdifferential equations can be traced back to the pioneering paper of Dziuk [15] onthe Laplace-Beltrami equation. Meanwhile there are various extensions to movinghypersurfaces such as, e.g., evolving surface finite element methods [16, 17] or tracefinite element methods [36], and an abstract framework for parabolic equations onevolving Hilbert spaces [1, 2].Though uncertain parameters are rather the rule than the exception in manyapplications and though partial differential equations with random coefficients havebeen intensively studied over the last years (cf., e.g., the monographs [31] and [29]),the numerical analysis of random surface partial differential equations still appearsto be in its infancy.In this paper, we present random evolving surface finite element methods for theadvection-diffusion equation ∂ • u − ∇ Γ ( α ∇ Γ u ) + u ∇ Γ · v = f on an evolving compact hypersurface Γ( t ) ⊂ R n , n = 2, 3, with a uniformlybounded random coefficient α and deterministic velocity v on a compact timeintervall t ∈ [0 , T ]. Here ∂ • denotes the path-wise material derivative and ∇ Γ is the tangential gradient. While the analysis and numerical analysis of randomadvection-diffusion equations is well developed in the flat case [8, 25, 28, 33], to Mathematics Subject Classification.
Key words and phrases. geometric partial differential equations, surface finite elements, ran-dom advection-diffusion equation, uncertainty quantification.The research of TR was funded by the Engineering and Physical Sciences Research Council(EPSRC EP/J004057/1). The research of CME was partially supported by the Royal Society viaa Wolfson Research Merit Award and by the EPSRC programme grant (EP/K034154/1) EQUIP. a r X i v : . [ m a t h . NA ] S e p A. DJURDJEVAC, C. M. ELLIOTT, R. KORNHUBER, AND T. RANNER our knowledge, existence, uniqueness and regularity results for curved domainshave been first derived only recently in [14]. Following Dziuk & Elliott [16], thespace discretization is performed by random piecewise linear finite element func-tions on simplicial approximations Γ h ( t ) of the surface Γ( t ), t ∈ [0 , T ]. We presentoptimal error estimates for the resulting semi-discrete scheme which then providecorresponding error estimates for expectation values and Monte-Carlo approxima-tions. Application of efficient solution techniques, such as adaptivity [13], multigridmethods [27], and Multilevel Monte-Carlo techniques [3, 9, 10] is very promisingbut beyond the scope of this paper. In our numerical experiments we investigate acorresponding fully discrete scheme based on an implicit Euler method and observeoptimal convergence rates.The paper is organized as follows. We start by setting up some notation, thenotion of hypersurfaces, function spaces, and material derivatives in order to de-rive a weak formulation of our problem according to [14]. Section 3 is devotedto the random ESFEM discretization in the spirit of [16] leading to the preciseformulation and well-posedness of our semi discretization in space presented in Sec-tion 4. Optimal error estimates for the approximate solution, its expectation anda Monte-Carlo approximation are contained in Section 5. The paper concludeswith numerical experiments in two and three space dimensions suggesting that ouroptimal error estimates extend to corresponding fully discrete schemes.2. Random advection-diffusion equations on evolving hypersurfaces
Let (Ω , F , P ) be a complete probability space with sample space Ω, a σ -algebraof events F and a probability P : F → [0 , L (Ω) is aseparable space. For this assumption it suffices to assume that (Ω , F , P ) is separable[23, Exercise 43.(1)]. We consider a fixed finite time interval [0 , T ], where T ∈ (0 , ∞ ) . Furthermore, we denote by D ((0 , T ); V ) the space of infinitely differentiablefunctions with values in a a Hilbert space V and compact support in (0 , T ).2.1. Hypersurfaces.
We first recall some basic notions and results concerninghypersurfaces and Sobolev spaces on hypersurfaces. We refer to [11] and [19] formore details.Let Γ ⊂ R n +1 ( n = 1 ,
2) be a C -compact, connected, orientable, n -dimensionalhypersurface without boundary. For a function f : Γ → R allowing for a differen-tiable extension ˜ f to an open neighbourhood of Γ in R n +1 we define the tangentialgradient by(2.1) ∇ Γ f ( x ) := ∇ ˜ f ( x ) − ∇ ˜ f ( x ) · ν ( x ) ν ( x ) , x ∈ Γ , where ν ( x ) denotes the unit normal to Γ.Note that ∇ Γ f ( x ) is the orthogonal projection of ∇ ˜ f onto the tangent spaceto Γ at x (thus a tangential vector). It depends only on the values of ˜ f on Γ[19, Lemma 2.4], which makes the definition (2.1) independent of the extension ˜ f .The tangential gradient is a vector-valued quantity and for its components we usethe notation ∇ Γ f ( x ) = ( D f ( x ) , . . . , D n +1 f ( x )) . The
Laplace-Beltrami operator isdefined by ∆ Γ f ( x ) = ∇ Γ · ∇ Γ f ( x ) = n +1 (cid:88) i =1 D i D i f ( x ) , x ∈ Γ . SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 3
In order to prepare weak formulations of PDEs on Γ, we now introduce Sobolevspaces on surfaces. To this end, let L (Γ) denote the Hilbert space of all measurablefunctions f : Γ → R such that (cid:107) f (cid:107) L (Γ) := (cid:0)(cid:82) Γ | f ( x ) | (cid:1) / is finite. We say that afunction f ∈ L (Γ) has a weak partial derivative g i = D i f ∈ L (Γ) , ( i = { , . . . , n +1 } ), if for every function φ ∈ C (Γ) and every i there holds (cid:90) Γ f D i φ = − (cid:90) Γ φg i + (cid:90) Γ f φHν i where H = −∇ Γ · ν denotes the mean curvature. The Sobolev space H (Γ) is thendefined by H (Γ) = { f ∈ L (Γ) | D i f ∈ L (Γ) , i = 1 , . . . , n + 1 } with the norm (cid:107) f (cid:107) H (Γ) = ( (cid:107) f (cid:107) L (Γ) + (cid:107)∇ Γ f (cid:107) L (Γ) ) / .For a description of evolving hypersurfaces we consider two approaches, startingwith evolutions according to a given velocity field v. Here, we assume that Γ( t )satisfies the same properties as Γ(0) = Γ for every t ∈ [0 , T ], and we set Γ := Γ(0).Furthermore, we assume the existence of a flow, i.e., of a diffeomorphismΦ t ( · ) := Φ( · , t ) : Γ → Γ( t ) , Φ ∈ C ([0 , T ] , C (Γ ) n +1 ) ∩ C ([0 , T ] , C (Γ ) n +1 ) , that satisfies(2.2) ddt Φ t ( · ) = v( t, Φ t ( · )) , Φ ( · ) = Id( · ) , with a C -velocity field v : [0 , T ] × R n +1 → R n +1 with uniformly bounded divergence(2.3) |∇ Γ( t ) · v( t ) | ≤ C ∀ t ∈ [0 , T ] . It is sometimes convenient to alternatively represent Γ( t ) as the zero level set ofa suitable function defined on a subset of the ambient space R n +1 . More precisely,under the given regularity assumptions for Γ( t ), it follows by the Jordan-Brouwertheorem that Γ( t ) is the boundary of an open bounded domain. Thus, Γ( t ) can berepresented as the zero level setΓ( t ) = { x ∈ N ( t ) | d ( x, t ) = 0 } , t ∈ [0 , T ] , of a signed distance function d = d ( x, t ) defined on an open neighborhood N ( t ) ofΓ( t ) such that |∇ d | (cid:54) = 0 for t ∈ [0 , T ]. Note that d , d t , d x i , d x i x j ∈ C ( N T ) with i , j = 1 , . . . , n + 1 holds for N T := (cid:91) t ∈ [0 ,T ] N ( t ) × { t } . We also choose N ( t ) such that for every x ∈ N ( t ) and t ∈ [0 , T ] there exists aunique p ( x, t ) ∈ Γ( t ) such that(2.4) x = p ( x, t ) + d ( x, t ) ν ( p ( x, t ) , t ) , and fix the orientation of Γ( t ) by choosing the normal vector field ν ( x, t ) := ∇ d ( x, t ).Note that the constant extension of a function η ( · , t ) : Γ( t ) → R to N ( t ) in normaldirection is given by η − l ( x, t ) = η ( p ( x, t ) , t ), p ∈ N ( t ). Later on, we will use (2.4)to define the lift of functions on approximate hypersurfaces. A. DJURDJEVAC, C. M. ELLIOTT, R. KORNHUBER, AND T. RANNER
Function spaces.
In this section, we define Bochner-type function spaces ofrandom functions that are defined on evolving spaces. The definition of these spacesis taken from [14] and uses the idea from Alphonse et al. [1] to map each domain attime t to the fixed initial domain Γ by a pull-back operator using the flow Φ t . Notethat this approach is similar to Arbitrary Lagrangian Eulerian (ALE) framework.For each t ∈ [0 , T ], let us define V ( t ) := L (Ω , H (Γ( t ))) ∼ = L (Ω) ⊗ H (Γ( t ))(2.5) H ( t ) := L (Ω , L (Γ( t ))) ∼ = L (Ω) ⊗ L (Γ( t ))(2.6)where the isomorphisms hold because all considered spaces are separable Hilbertspaces (see [35]). The dual space of V ( t ) is the space V ∗ ( t ) = L (Ω , H − (Γ( t ))),where H − (Γ( t )) is the dual space of H (Γ( t )). Using the tensor product structureof these spaces [22, Lemma 4.34], it follows that V ( t ) ⊂ H ( t ) ⊂ V ∗ ( t ) is a Gelfandtriple for every t ∈ [0 , T ]. For convenience we will often (but not always) write u ( ω, x ) instead of u ( ω )( x ), which is justified by the tensor structure of the spaces.For an evolving family of Hilbert spaces X = ( X ( t )) t ∈ [0 ,T ] , such as, e.g., V =( V ( t )) t ∈ [0 ,T ] or H = ( H ( t )) t ∈ [0 ,T ] we connect the space X ( t ) for fixed t ∈ [0 , T ] withthe initial space X (0) by using a family of so-called pushforward maps φ t : X (0) → X ( t ), satisfying certain compatibility conditions stated in [1, Definition 2.4]. Moreprecisely, we use its inverse map φ − t : X ( t ) → X (0), called pullback map, to definegeneral Bochner-type spaces of functions defined on evolving spaces as follows (see[1, 14]) L X := u : [0 , T ] (cid:51) t (cid:55)→ (¯ u ( t ) , t ) ∈ (cid:91) t ∈ [0 ,T ] X ( t ) × { t } | φ − ( · ) ¯ u ( · ) ∈ L (0 , T ; X (0)) ,L X ∗ := f : [0 , T ] (cid:51) t (cid:55)→ ( ¯ f ( t ) , t ) ∈ (cid:91) t ∈ [0 ,T ] X ∗ ( t ) × { t } | φ − ( · ) ¯ f ( · ) ∈ L (0 , T ; X (0) ∗ ) . In the following we will identify u ( t ) = ( u ( t ); t ) with u ( t ).From [1, Lemma 2.15] it follows that L X ∗ and ( L X ) ∗ are isometrically isomor-phic. The spaces L X and L X ∗ are separable Hilbert spaces [1, Corollary 2.11] withthe inner product defined as( u, v ) L X = (cid:90) T ( u ( t ) , v ( t )) X ( t ) d t ( f, g ) L X ∗ = (cid:90) T ( f ( t ) , g ( t )) X ∗ ( t ) d t. For the evolving family H defined in (2.6) we define the pullback operator φ − t : H ( t ) → H (0) for fixed t ∈ [0 , T ] and each u ∈ H ( t ) by( φ − t u )( ω, x ) := u ( ω, Φ t ( x )) , x ∈ Γ = Γ(0) , ω ∈ Ω , utilizing the parametrisation Φ t of Γ( t ) over Γ . Exploiting V ( t ) ⊂ H ( t ), thepullback operator φ − t : V ( t ) → V (0) is defined by restriction. It follows from [14,Lemma 3.5] that the resulting spaces L V , L V ∗ and L H are well-defined and L V ⊂ L H ⊂ L V ∗ is a Gelfand triple. SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 5
Material derivative.
Following [14], we introduce a material derivative ofsufficiently smooth random functions that takes spatial movement into account.First let us define the spaces of pushed-forward continuously differentiable func-tions C jX := { u ∈ L X | φ − ( · ) u ( · ) ∈ C j ([0 , T ] , X (0)) } for j ∈ { , , } . For u ∈ C V the material derivative ∂ • u ∈ C V is defined by(2.7) ∂ • u := φ t (cid:18) ddt φ − t u (cid:19) = u t + ∇ u · v . More precisely, the material derivative of u is defined via a smooth extension ˜ u of u to N T with well-defined derivatives ∇ ˜ u and ˜ u t and subsequent restriction to G T := (cid:91) t Γ( t ) × { t } ⊂ N T . Since, due to the smoothness of Γ( t ) and Φ t , this definition is independent of thechoice of particular extension ˜ u , we simply write u in (2.7). Remark 2.1.
Replacing classical derivatives in time by weak derivatives leads to aweak material derivative ∂ • u ∈ L V ∗ . It coincides with the strong material derivativefor sufficiently smooth functions. As we will concentrate on the smooth case lateron, we omit a precise definition here and refer to [14, Definition 3.9] for details. Weak formulation and well-posedness.
We consider an initial value prob-lem for an advection-diffusion equation on the evolving surface Γ( t ), t ∈ [0 , T ], whichin strong form reads(2.8) ∂ • u − ∇ Γ · ( α ∇ Γ u ) + u ∇ Γ · v = fu (0) = u . Here the diffusion coefficient α and the initial function u are random functions,and we set f ≡ W ( V, H ) := { u ∈ L V | ∂ • u ∈ L H } where ∂ • u stands for the weak material derivative. W ( V, H ) is a separable Hilbertspace with the inner product defined by( u, v ) W ( V,H ) = (cid:90) T (cid:90) Ω ( u, v ) H (Γ( t )) + (cid:90) T (cid:90) Ω ( ∂ • u, ∂ • v ) L (Γ( t )) . Now a a weak solution of (2.8) is a solution of the following problem.
Problem 2.1 (Weak form of the random advection-diffusion equation on { Γ( t ) } ) . Find u ∈ W ( V, H ) that point-wise satisfies the initial condition u (0) = u ∈ V (0) and (2.10) (cid:90) Ω (cid:90) Γ( t ) ∂ • u ( t ) ϕ + (cid:90) Ω (cid:90) Γ( t ) α ( t ) ∇ Γ u ( t ) · ∇ Γ ϕ + (cid:90) Ω (cid:90) Γ( t ) u ( t ) ϕ ∇ Γ · v( t ) = 0 , for every ϕ ∈ L (Ω , H (Γ( t ))) and a.e. t ∈ [0 , T ] . Existence and uniqueness can be stated on the following assumption.
Assumption 2.1.
The diffusion coefficient α satisfies the following conditions a) α : Ω × G T → R is a F ⊗ B ( G T ) -measurable. A. DJURDJEVAC, C. M. ELLIOTT, R. KORNHUBER, AND T. RANNER b) α ( ω, · , · ) ∈ C ( G T ) holds for P -a.e ω ∈ Ω , which implies boundedness of | ∂ • α ( ω ) | on G T , and we assume that this bound is uniform in ω ∈ Ω . c) α is uniformly bounded from above and below in the sense that there existpositive constants α min and α max such that (2.11) 0 < α min ≤ α ( ω, x, t ) ≤ α max < ∞ ∀ ( x, t ) ∈ G T holds for P -a.e. ω ∈ Ω and the initial function satisfies u ∈ L (Ω , H (Γ )) . The following proposition is a consequence of [14, Theorem 4.9].
Proposition 2.1.
Let Assumption 2.1 hold. Then, under the given assumptionson { Γ( t ) } , there is a unique solution u ∈ W ( V, H ) of Problem 2.1 and we have thea priori bound (cid:107) u (cid:107) W ( V,H ) ≤ C (cid:107) u (cid:107) V (0) with some C ∈ R . The following assumption of the diffusion coefficient will ensure regularity of thesolution.
Assumption 2.2.
Assume that there exists a constant C independent of ω ∈ Ω such that |∇ Γ α ( ω, x, t ) | ≤ C ∀ ( x, t ) ∈ G T holds for P -almost all ω ∈ Ω . Note that (2.11) and Assumption 2.2 imply that (cid:107) α ( ω, t ) (cid:107) C (Γ( t )) is uniformlybounded in ω ∈ Ω. This will be used later to prove an H (Γ( t )) bound. In thesubsequent error analysis, we will assume further that u has a path-wise strongmaterial derivative, i.e. that u ( ω ) ∈ C V holds for all ω ∈ Ω.In order to derive a more convenient formulation of Problem 2.1 with identicalsolution and test space, we introduce the time dependent bilinear forms(2.12) m ( u, ϕ ) := (cid:90) Ω (cid:90) Γ( t ) uϕ, g (v; u, ϕ ) := (cid:90) Ω (cid:90) Γ( t ) uϕ ∇ Γ · v ,a ( u, ϕ ) := (cid:90) Ω (cid:90) Γ( t ) α ∇ Γ u · ∇ Γ ϕ, b (v; u, ϕ ) := (cid:90) Ω (cid:90) Γ( t ) B ( ω, v) ∇ Γ u · ∇ Γ ϕ for u, ϕ ∈ L (Ω , H (Γ( t ))) and each t ∈ [0 , T ]. The tensor B in the definition of b (v; u, ϕ ) takes the form B ( ω, v) = ( ∂ • α + α ∇ Γ · v)Id − αD Γ (v)with Id denoting the identity in ( n + 1) × ( n + 1) and ( D Γ v) ij = D j v i . Note that(2.3) and the uniform boundedness of ∂ • α on G T imply that | B ( ω, v) | ≤ C holds P -a.e. ω ∈ Ω with some C ∈ R .The transport formula for the differentiation of the time dependent surface in-tegral then reads (see e.g. [14]) ddt m ( u, ϕ ) = m ( ∂ • u, ϕ ) + m ( u, ∂ • ϕ ) + g (v; u, ϕ ) , (2.13)where the equality holds a.e. in [0 , T ]. As a consequence of (2.13), Problem 2.1 isequivalent to the following formulation with identical solution and test space. SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 7
Problem 2.2 (Weak form of the random advection-diffusion equation on { Γ( t ) } ) . Find u ∈ W ( V, H ) that point-wise satisfies the initial condition u (0) = u ∈ V (0) and (2.14) ddt m ( u, ϕ ) + a ( u, ϕ ) = m ( u, ∂ • ϕ ) ∀ ϕ ∈ W ( V, H ) . This formulation will be used in the sequel.3.
Evolving simplicial surfaces
As a first step towards a discretization of the weak formulation (2.14) we nowconsider simplicial approximations of the evolving surface Γ( t ), t ∈ [0 , T ]. Let Γ h, be an approximation of Γ consisting of nondegenerate simplices { E j, } Nj =1 =: T h, with vertices { X j, } Jj =1 ⊂ Γ such that the intersection of two different simplicesis a common lower dimensional simplex or empty. For t ∈ [0 , T ], we let the ver-tices X j (0) = X j, evolve with the smooth surface velocity X (cid:48) j ( t ) = v( X j ( t ) , t ), j = 1 , . . . , J , and consider the approximation Γ h ( t ) of Γ( t ) consisting of the corre-sponding simplices { E j ( t ) } Mj =1 =: T h ( t ). We assume that shape regularity of T h ( t )holds uniformly in t ∈ [0 , T ] and that T h ( t ) is quasi-uniform, uniformly in time, inthe sense that h := sup t ∈ (0 ,T ) max E ( t ) ∈T h ( t ) diam E ( t ) ≥ inf t ∈ (0 ,T ) min E ( t ) ∈T h ( t ) diam E ( t ) ≥ ch holds with some c ∈ R . We also assume that Γ h ( t ) ⊂ N ( t ) for t ∈ [0 , T ] and, inaddition to (2.4), that for every p ∈ Γ( t ) there is a unique x ( p, t ) ∈ Γ h ( t ) such that(3.1) p = x ( p, t ) + d ( x ( p, t ) , t ) ν ( p, t ) . Note that Γ h ( t ) can be considered as interpolation of Γ( t ) in { X j ( t ) } Jj =1 and adiscrete analogue of the space time domain G T is given by G hT := (cid:91) t Γ h ( t ) × { t } . We define the tangential gradient of a sufficiently smooth function η h : Γ h ( t ) → R in an element-wise sense, i.e., we set ∇ Γ h η h | E = ∇ η h − ∇ η h · ν h ν h , E ∈ T h ( t ) . Here ν h stands for the element-wise outward unit normal to E ⊂ Γ h ( t ). We use thenotation ∇ Γ h η h = ( D h, η h , . . . , D h,n +1 η h ).We define the discrete velocity V h of Γ h ( t ) by interpolation of the given velocity v,i.e. we set V h ( X ( t ) , t ) := ˜ I h v( X ( t ) , t ) , X ( t ) ∈ Γ h ( t ) , with ˜ I h denoting piecewise linear interpolation in { X j ( t ) } Jj =1 .We consider the Gelfand triple on Γ h ( t )(3.2) L (Ω , H (Γ h ( t ))) ⊂ L (Ω , L (Γ h ( t ))) ⊂ L (Ω , H − (Γ h ( t )))and denote V h ( t ) := L (Ω , H (Γ h ( t ))) and H h ( t ) := L (Ω , L (Γ h ( t ))) . As in the continuous case, this leads to the following Gelfand triple of evolvingBochner-Sobolev spaces(3.3) L V h ( t ) ⊂ L H h ( t ) ⊂ L V ∗ h ( t ) . A. DJURDJEVAC, C. M. ELLIOTT, R. KORNHUBER, AND T. RANNER
The discrete velocity V h induces a discrete strong material derivative in termsof an element-wise version of (2.7), i.e., for sufficiently smooth functions φ h ∈ L V h and any E ( t ) ∈ Γ h ( t ) we set(3.4) ∂ • h φ h | E ( t ) := ( φ h,t + V h · ∇ φ h ) | E ( t ) . We define discrete analogues to the bilinear forms introduced in (2.12) on V h ( t ) ×V h ( t ) according to m h ( u h , ϕ h ) := (cid:90) Ω (cid:90) Γ h ( t ) u h ϕ h , g h ( V h ; u h , ϕ h ) := (cid:90) Ω (cid:90) Γ h ( t ) u h ϕ h ∇ Γ h · V h ,a h ( u h , ϕ h ) := (cid:90) Ω (cid:90) Γ h ( t ) α − l ∇ Γ h u h · ∇ Γ h ϕ h ,b h ( V h ; φ, U h ) := (cid:88) E ( t ) ∈T h ( t ) (cid:90) Ω (cid:90) E ( t ) B h ( ω, V h ) ∇ Γ h φ · ∇ Γ h U h involving the tensor B h ( ω, V h ) = ( ∂ • h α − l + α − l ∇ Γ h · V h )Id − α − l D h ( V h )denoting ( D h ( V h )) ij = D h,j V ih . Here, we denote(3.5) α − l ( ω, x, t ) := α ( ω, p ( x, t ) , t ) ω ∈ Ω , ( x, t ) ∈ G hT exploiting { Γ h ( t ) } ⊂ N ( t ) and (2.4). Later α − l will be called the inverse lift of α .Note that α − l satisfies a discrete version of Assumption 2.1 and 2.2. In particular, α − l is an F ⊗ B ( G hT )-measurable function, α − l ( ω, · , · ) | E T ∈ C ( E T ) for all space-time elements E T := (cid:83) t E ( t ) × { t } , and α min ≤ α − l ( ω, x, t ) ≤ α max for all ω ∈ Ω,( x, t ) ∈ G hT .The next lemma provides a uniform bound for the divergence of V h and the normof the tensor B h that follows from the geometric properites of Γ h ( t ) in analogy to[20, Lemma 3.3]. Lemma 3.1.
Under the above assumptions on { Γ h ( t ) } , it holds sup t ∈ [0 ,T ] (cid:0) (cid:107)∇ Γ h · V h (cid:107) L ∞ (Γ h ( t )) + (cid:107) B h (cid:107) L (Ω ,L ∞ (Γ h ( t ))) (cid:1) ≤ c sup t ∈ [0 ,T ] (cid:107) v( t ) (cid:107) C ( N T ) with a constant c depending only on the initial hypersurface Γ and the uniformshape regularity and quasi-uniformity of T h ( t ) . Since the probability space does not depend on time, the discrete analogue ofthe corresponding transport formulae hold, where the discrete material velocityand discrete tangential gradients are understood in an element-wise sense. Theresulting discrete result is stated for example in [17, Lemma 4.2]. The followinglemma follows by integration over Ω.
Lemma 3.2 (Transport lemma for triangulated surfaces) . Let { Γ h ( t ) } be a fam-ily of triangulated surfaces evolving with discrete velocity V h . Let φ h , η h be timedependent functions such that the following quantities exist. Then ddt (cid:90) Ω (cid:90) Γ h ( t ) φ h = (cid:90) Ω (cid:90) Γ h ( t ) ∂ • h φ h + φ h ∇ Γ h · V h . In particular, (3.6) ddt m h ( φ h , η h ) = m ( ∂ • h φ h , η h ) + m ( φ h , ∂ • h η h ) + g h ( V h ; φ h , η h ) . SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 9 Evolving surface finite element methods
Following [16], we now introduce an evolving surface finite element discretization(ESFEM) of Problem 2.2.4.1.
Finite elements on simplicial surfaces.
For each t ∈ [0 , T ] we define the evolving finite element space (4.1) S h ( t ) := { η ∈ C (Γ h ( t )) | η E is affine ∀ E ∈ T h ( t ) } . We denote by { χ j ( t ) } j =1 ,...,J the nodal basis of S h ( t ), i.e. χ j ( X i ( t ) , t ) = δ ij (Kronecker- δ ). These basis functions satisfy the transport property [17, Lemma4.1](4.2) ∂ • h χ j = 0 . We consider the following Gelfand triple(4.3) S h ( t ) ⊂ L h ( t ) ⊂ S ∗ h ( t ) , where all three spaces algebraically coincide but are equipped with different normsinherited from the corresponding continuous counterparts, i.e., S h ( t ) := ( S h ( t ) , (cid:107) · (cid:107) H (Γ h ( t )) ) and L h ( t ) := ( S h ( t ) , (cid:107) · (cid:107) L (Γ h ( t )) ) . The dual space S ∗ h ( t ) consists of all continuous linear functionals on S h ( t ) and isequipped with the standard dual norm (cid:107) ψ (cid:107) S ∗ h ( t ) := sup { η ∈ S h ( t ) | (cid:107) η (cid:107) H h ( t )) =1 } | ψ ( η ) | . Note that all three norms are equivalent as norms on finite dimensional spaces,which implies that (4.3) is the Gelfand triple. As a discrete counterpart of (3.2),we introduce the Gelfand triple(4.4) L (Ω , S h ( t )) ⊂ L (Ω , L h ( t )) ⊂ L (Ω , S ∗ h ( t )) . Setting V h ( t ) := L (Ω , S h ( t )) H h ( t ) := L (Ω , L h ( t )) V ∗ h ( t ) := L (Ω , S ∗ h ( t ))we obtain the finite element analogue(4.5) L V h ( t ) ⊂ L H h ( t ) ⊂ L V ∗ h ( t ) of the Gelfand triple (3.3) of evolving Bochner-Sobolev spaces. Let us note thatsince the sample space Ω is independent of time, it holds(4.6) L (Ω , L X ) ∼ = L (Ω) ⊗ L X ∼ = L L (Ω ,X ) for any evolving family of separable Hilbert spaces X (see, e.g., Section 3). Wewill exploit this isomorphism for X = S h in the following definition of the solutionspace for the semi-discrete problem, where we will rather consider the problem ina path-wise sense.We define the solution space for the semi-discrete problem as the space of func-tions that are smooth for each path in the sense that φ h ( ω ) ∈ C S h holds for all ω ∈ Ω. Hence, ∂ • h φ h is defined path-wise for path-wise smooth functions. In addi-tion, we require ∂ • h φ h ( t ) ∈ H h ( t ) to define the semi-discrete solution space W h ( V h , H h ) := L (Ω , C S h ) . The scalar product of this space is defined by( U h , φ h ) W h ( V h ,H h ) := (cid:90) T (cid:90) Ω ( U h , φ h ) H (Γ h ( t )) + (cid:90) T (cid:90) Ω ( ∂ • h U h , ∂ • h φ h ) L (Γ h ( t )) with the associated norm (cid:107) · (cid:107) W h ( V h ,H h ) .The semi-discrete approximation of Problem 2.2, on { Γ h ( t ) } now reads as follows. Problem 4.1 (ESFEM discretization in space) . Find U h ∈ W h ( V h , H h ) that point-wise satisfies the initial condition U h (0) = U h, ∈ V h (0) and (4.7) ddt m h ( U h , ϕ ) + a h ( U h , ϕ ) = m h ( U h , ∂ • h ϕ ) ∀ ϕ ∈ W h ( V h , H h ) . In contrast to W ( V, H ), the semidiscrete space W h ( V h , H h ) is not complete sothat the proof of the following existence and stability result requires a different kindof argument. Theorem 4.1.
The semi-discrete problem (4.9) has a unique solution U h ∈ W h ( V h , H h ) which satisfies the stability property (4.8) (cid:107) U h (cid:107) W ( V h ,H h ) ≤ C (cid:107) U h, (cid:107) V h (0) with a mesh-independent constant C depending only on T , α min , and the bound for (cid:107)∇ Γ h · V h (cid:107) ∞ from Lemma 3.1.Proof. In analogy to Subsection 2.4, Problem 4.1 is equivalent to find U h ∈ W h ( V h , H h )that point-wise satisfies the initial condition U h (0) = U h, ∈ V h (0) and(4.9) m h ( ∂ • h U h , ϕ ) + a ( U h , ϕ ) + g ( V h ; U h , ϕ ) = 0for every ϕ ∈ L (Ω , S h ( t )) and a.e. t ∈ [0 , T ].Let ω ∈ Ω be arbitrary but fixed. We start with considering the deterministicpath-wise problem to find U h ( ω ) ∈ C S h such that U h ( ω ; 0) = U h, ( ω ) and(4.10) (cid:90) Γ h ( t ) ∂ • h U h ( ω ) ϕ + (cid:90) Γ h ( t ) α − l ( ω ) ∇ Γ h U h ( ω ) · ∇ Γ h ϕ + (cid:90) Γ h ( t ) U h ( ω ) ϕ ∇ Γ h · V h = 0holds for all ϕ ∈ S h ( t ) and a.e. t ∈ [0 , T ]. Following Dziuk & Elliott [17, Section4.6], we insert the nodal basis representation(4.11) U h ( ω, t, x ) = J (cid:88) j =1 U j ( ω, t ) χ j ( x, t )into (4.10) and take ϕ = χ i ( t ) ∈ S h ( t ) , i = 1 , . . . , J , as test functions. Now thetransport property (4.2) implies J (cid:88) j =1 ∂∂t U j ( ω ) (cid:90) Γ h ( t ) χ j χ i + J (cid:88) j =1 U j ( ω ) (cid:90) Γ h ( t ) α − l ( ω ) ∇ Γ h χ j · ∇ Γ h χ i (4.12) + J (cid:88) j =1 U j ( ω ) (cid:90) Γ h ( t ) χ j χ i ∇ Γ h · V h = 0 . We introduce the evolving mass matrix M ( t ) with coefficients M ( t ) ij := (cid:90) Γ h ( t ) χ i ( t ) χ j ( t ) , SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 11 and the evolving stiffness matrix S ( ω, t ) with coefficients S ( ω, t ) ij := (cid:90) Γ h ( t ) α − l ( ω, t ) ∇ Γ h χ j ( t ) ∇ Γ h χ i ( t ) . From [17, Proposition 5.2] it follows dMdt = M (cid:48) where M (cid:48) ( t ) ij := (cid:90) Γ h ( t ) χ j ( t ) χ i ( t ) ∇ Γ h · V h ( t ) . Therefore, we can write (4.12) as the following linear initial value problem(4.13) ∂∂t ( M ( t ) U ( ω, t )) + S ( ω, t ) U ( ω, t ) = 0 , U ( ω,
0) = U ( ω ) , for the unknown vector U ( ω, t ) = ( U j ( ω, t )) Ji =1 of coefficient functions. As in [17],there exists an unique path-wise semi-discrete solution U h ( ω ) ∈ C S h , since thematrix M ( t ) is uniformly positive definite on [0 , T ] and the stiffness matrix S ( ω, t )is positive semi-definite for every ω ∈ Ω. Note that the time regularity of U h ( ω )follows from M , S ( ω ) ∈ C (0 , T ) which in turn is a consequence of our assumptionson the time regularity of the evolution of Γ h ( t ).The next step is to prove the measurability of the map Ω (cid:51) ω (cid:55)→ U h ( ω ) ∈ C S h .On C S h we consider the Borel σ − algebra induced by the norm(4.14) (cid:107) U h (cid:107) C Sh := (cid:90) T (cid:107) U h ( t ) (cid:107) H (Γ h ( t )) + (cid:107) ∂ • h U h ( t ) (cid:107) L (Γ h ( t )) . We write (4.12) in the following form ∂∂t U ( ω, t ) + A ( ω, t ) U ( ω, t ) = 0 , U ( ω,
0) = U ( ω ) , where A ( ω, t ) := M − ( t ) ( M (cid:48) ( t ) + S ( ω, t )) . As U h, ∈ V h (0), the function ω (cid:55)→ U ( ω ) is measurable and since α − l is a F ⊗ B ( G hT )-measurable function, it follows from Fubini’s Theorem [23, Sec. 36,Thm. C] thatΩ (cid:51) ω (cid:55)→ ( U ( ω ) , A ( ω )) ∈ R J × (cid:0) C (cid:0) [0 , T ] , R J (cid:1) , (cid:107) · (cid:107) ∞ (cid:1) is measurable function. Utilizing Gronwall’s lemma it can be shown that the map-ping R J × (cid:0) C (cid:0) [0 , T ] , R J (cid:1) , (cid:107) · (cid:107) ∞ (cid:1) (cid:51) ( U , A ) (cid:55)→ U ∈ (cid:0) C (cid:0) [0 , T ] , R J (cid:1) , (cid:107) · (cid:107) ∞ (cid:1) is continuous. Furthermore, the mapping (cid:0) C (cid:0) [0 , T ] , R J (cid:1) , (cid:107) · (cid:107) ∞ (cid:1) (cid:51) U (cid:55)→ U ∈ (cid:0) C (cid:0) [0 , T ] , R J (cid:1) , (cid:107) · (cid:107) (cid:1) with (cid:107) U (cid:107) := (cid:90) T (cid:107) U ( t ) (cid:107) R J + (cid:107) ddt U ( t ) (cid:107) R J is continuous. Exploiting that the triangulation T h ( t ) of Γ h ( t ) is quasi-uniform,uniformly in time, the continuity of the linear mapping (cid:0) C (cid:0) [0 , T ] , R J (cid:1) , (cid:107) · (cid:107) (cid:1) (cid:51) U (cid:55)→ U h ∈ C S h follows from the triangle inequality and the Cauchy-Schwarz inequality. We finallyconclude that the function Ω (cid:51) ω (cid:55)→ U h ( ω ) ∈ C S h is measurable as a composition of measurable and continuous mappings.The next step is to prove the stability property (4.8). For each fixed ω ∈ Ω,path-wise stability results from [17, Lemma 4.3] imply(4.15) (cid:107) U h ( ω ) (cid:107) C Sh ≤ C (cid:107) U h, ( ω ) (cid:107) H (Γ h (0)) where C = C ( α min , α max , V h , T, G Th ) is independent of ω and U h, ( x ) ∈ L (Ω).Integrating (4.15) over Ω we get the bound (cid:107) U h (cid:107) W ( V h ,H h ) = (cid:107) U h (cid:107) L (Ω , C Sh ) ≤ C (cid:107) U h, (cid:107) V h (0) . In particular, we have U h ∈ W h ( V h , H h ).It is left to show that U h solves (4.9) and thus Problem 4.1. Exploiting thetensor product structure of the test space L (Ω , S h ( t )) ∼ = L (Ω) ⊗ S h ( t ) (see (4.6)),we find that { ϕ h ( x, t ) η ( ω ) | ϕ h ( t ) ∈ S h ( t ) , η ∈ L (Ω) } ⊂ L (Ω) ⊗ S h ( t )is a dense subset of L (Ω , S h ( t )). Taking any test function ϕ h ( x, t ) η ( ω ) from thisdense subset, we first insert ϕ h ( x, t ) ∈ S h ( t ) into the pathwise problem (4.10), thenmultiply with η ( ω ), and finally integrate over Ω to establish (4.9). This completesthe proof. (cid:3) Lifted finite elements.
We exploit (3.1) to define the lift η lh ( · , t ) : Γ( t ) → R of functions η h ( · , t ) : Γ h ( t ) → R by η lh ( p, t ) := η h ( x ( p, t )) , p ∈ Γ( t ) . Conversely, (2.4) is utilized to define the inverse lift η − l ( · , t ) : Γ h ( t ) → R of functions η ( · , t ) : Γ( t ) → R by η − l ( x, t ) := η ( p ( x, t ) , t ) , x ∈ Γ h ( t ) . These operators are inverse to each other, i.e., ( η − l ) l = ( η l ) − l = η , and, takingcharacteristic functions η h , each element E ( t ) ∈ T h ( t ) has its unique associatedlifted element e ( t ) ∈ T lh ( t ). Recall that the inverse lift α − of the diffusion coefficient α was already introduced in (3.5).The next lemma states equivalence relations between corresponding norms onΓ( t ) and Γ h ( t ) that follow directly from their deterministic counterparts (see [15]). Lemma 4.1.
Let t ∈ [0 , T ] , ω ∈ Ω , and let η h ( ω ) : Γ h ( t ) → R with the lift η lh ( ω ) : Γ → R . Then for each plane simplex E ⊂ Γ h ( t ) and its curvilinear lift e ⊂ Γ( t ) , there is a constant c > independent of E , h, t, and ω such that c (cid:107) η h (cid:107) L (Ω ,L ( E )) ≤ (cid:107) η lh (cid:107) L (Ω ,L ( e )) ≤ c (cid:107) η h (cid:107) L (Ω ,L ( E )) (4.16) 1 c (cid:107)∇ Γ h η h (cid:107) L (Ω ,L ( E )) ≤ (cid:107)∇ Γ η lh (cid:107) L (Ω ,L ( e )) ≤ c (cid:107)∇ Γ h η h (cid:107) L (Ω ,L ( E )) (4.17) 1 c (cid:107)∇ h η h (cid:107) L (Ω ,L ( E )) ≤ c (cid:107)∇ η lh (cid:107) L (Ω ,L ( e )) + ch (cid:107)∇ Γ η lh (cid:107) L (Ω ,L ( e )) , (4.18) if the corresponding norms are finite. SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 13
The motion of the vertices of the triangles E ( t ) ∈ {T h ( t ) } induces a discretevelocity v h of the surface { Γ( t ) } . More precisely, for a given trajectory X ( t ) ofa point on { Γ h ( t ) } with velocity V h ( X ( t ) , t ) the associated discrete velocity v h in Y ( t ) = p ( X ( t ) , t ) on Γ( t ) is defined by(4.19) v h ( Y ( t ) , t ) = Y (cid:48) ( t ) = ∂p∂t ( X ( t ) , t ) + V h ( X ( t ) , t ) · ∇ p ( X ( t ) , t ) . The discrete velocity v h gives rise to a discrete material derivative of functions ϕ ∈ L V in an element-wise sense, i.e., we set ∂ • h ϕ | e ( t ) := ( ϕ t + v h · ∇ ϕ ) | e ( t ) for all e ( t ) ∈ T lh ( t ), where ϕ t and ∇ ϕ are defined via a smooth extension, analogousto the definition (2.7).We introduce a lifted finite element space by S lh ( t ) := { η l ∈ C (Γ( t )) | η ∈ S h ( t ) } . Note that there is a unique correspondence between each element η ∈ S h ( t ) and η l ∈ S lh ( t ). Furthermore, one can show that for every φ h ∈ S h ( t ) here holds(4.20) ∂ • h ( φ lh ) = ( ∂ • h φ h ) l . Therefore, by (4.2) we get ∂ • h χ lj = 0 . We finally state an analogon to the transport Lemma 3.2 on simplicial surfaces.
Lemma 4.2. (Transport lemma for smooth triangulated surfaces.)Let Γ( t ) be an evolving surface decomposed into curved elements {T h ( t ) } whoseedges move with velocity v h . Then the following relations hold for functions ϕ h , u h such that the following quantities exist ddt (cid:90) Ω (cid:90) Γ( t ) ϕ h = (cid:90) Ω (cid:90) Γ( t ) ∂ • h ϕ h + ϕ h ∇ Γ · v h . and (4.21) ddt m ( ϕ, u h ) = m ( ∂ • h ϕ h , u h ) + m ( ϕ h , ∂ • h u h ) + g ( v h ; ϕ h , u h ) . Remark 4.1.
Let U h be the solution of the semi-discrete Problem 4.1 with initialcondition U h (0) = U h, and let u h = U lh with u h (0) = u h, = U lh, be its lift. Then,as a consequence of Theorem 4.1, (4.20) , and Lemma 4.1, the following estimate (4.22) (cid:107) u h (cid:107) W ( V,H ) ≤ C (cid:107) u h (0) (cid:107) V (0) holds with C depending on the constants C and c appearing in Theorem 4.1 andLemma 4.1, respectively. Error estimates
Interpolation and geometric error estimates.
In this section we formu-late the results concerning the approximation of the surface, which are in the de-terministic setting proved in [16] and [17]. Our goal is to prove that they still holdin the random case. The main task is to keep track of constants that appear andshow that they are independent of realization. This conclusion mainly follows fromthe assumption (2.11) about the uniform distribution of the diffusion coefficient.
Furthermore, we need to show that the extended definitions of the interpolationoperator and Ritz projection operator are integrable with respect to P .We start with an interpolation error estimate for functions η ∈ L (Ω , H (Γ( t ))),where the interpolation I h η is defined as the lift of piecewise linear nodal inter-polation (cid:101) I h η ∈ L (Ω , S h ( t )). Note that (cid:101) I h is well-defined, because the vertices( X j ( t )) Jj =1 of Γ h ( t ) lie on the smooth surface Γ( t ) and n = 2, 3. Lemma 5.1.
The interpolation error estimate (cid:107) η − I h η (cid:107) H ( t ) + h (cid:107)∇ Γ ( η − I h η ) (cid:107) H ( t ) ≤ ch (cid:0) (cid:107)∇ η (cid:107) H ( t ) + h (cid:107)∇ Γ η (cid:107) H ( t ) (cid:1) (5.1) holds for all η ∈ L (Ω , H (Γ( t ))) with a constant c depending only on the shaperegularity of Γ h ( t ) .Proof. The proof of the lemma follows directly from the deterministic case andLemma 4.1. (cid:3)
We continue with estimating the geometric perturbation errors in the bilinearforms.
Lemma 5.2.
Let t ∈ [0 , T ] be fixed. For W h ( · , t ) and φ h ( · , t ) ∈ L (Ω , S h ( t )) with corresponding lifts w h ( · , t ) and ϕ h ( · , t ) ∈ L (Ω , S lh ( t )) we have the followingestimates of the geometric error | m ( w h , ϕ h ) − m h ( W h , φ h ) | ≤ ch (cid:107) w h (cid:107) H ( t ) (cid:107) ϕ h (cid:107) H ( t ) (5.2) | a ( w h , ϕ h ) − a h ( W h , φ h ) | ≤ ch (cid:107)∇ Γ w h (cid:107) H ( t ) (cid:107)∇ Γ ϕ h (cid:107) H ( t ) (5.3) | g ( v h ; w h , ϕ h ) − g h ( V h ; W h , φ h ) | ≤ ch (cid:107) w h (cid:107) V ( t ) (cid:107) ϕ h (cid:107) V ( t ) (5.4) | m ( ∂ • h w h , ϕ h ) − m h ( ∂ • h W h , φ h ) | ≤ ch (cid:107) ∂ • h w h (cid:107) H ( t ) (cid:107) ϕ (cid:107) H ( t ) . (5.5) Proof.
The assertion follows from uniform bounds of α ( ω, t ) and ∂ • h α ( ω, t ) withrespect to ω ∈ Ω together with corresponding deterministic results obtained in [17]and [30]. (cid:3)
Since the velocity v of Γ( t ) is deterministic, we can use [17, Lemma 5.6] to controlits deviation from the discrete velocity v h on Γ( t ). Furthermore, [17, Corollary5.7] provides the following error estimates for the continuous and discrete materialderivative. Lemma 5.3.
For the continuous velocity v of Γ( t ) and the discrete velocity v h defined in (4.19) the estimate (5.6) | v − v h | + h |∇ Γ (v − v h ) | ≤ ch holds pointwise on Γ( t ) . Moreover, there holds (cid:107) ∂ • z − ∂ • h z (cid:107) H ( t ) ≤ ch (cid:107) z (cid:107) V ( t ) , z ∈ V ( t ) , (5.7) (cid:107)∇ Γ ( ∂ • z − ∂ • h z ) (cid:107) H ( t ) ≤ ch (cid:107) z (cid:107) L (Ω ,H (Γ)) , z ∈ L (Ω , H (Γ( t ))) , (5.8) provided that the left hand sides are well-defined. Remark 5.1.
Since v h is a C -velocity field by assumption, (5.6) implies a uniformupper bound for ∇ Γ( t ) · v h which in turn yields the estimate (5.9) | g (v h ; w, ϕ ) | ≤ c (cid:107) w (cid:107) H ( t ) (cid:107) ϕ (cid:107) H ( t ) , ∀ w, ϕ ∈ H ( t ) with a constant c independent of h . SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 15
Ritz projection.
For each fixed t ∈ [0 , T ] and β ∈ L ∞ (Γ( t )) with 0 < β min ≤ β ( x ) ≤ β max < ∞ a.e. on Γ( t ) the Ritz projection H (Γ( t )) (cid:51) v (cid:55)→ R β v ∈ S lh ( t )is well-defined by the conditions (cid:82) Γ( t ) R β v = 0 and(5.10) (cid:90) Γ( t ) β ∇ Γ R β v · ∇ Γ ϕ h = (cid:90) Γ( t ) β ∇ Γ v · ∇ Γ ϕ h ∀ ϕ h ∈ S lh ( t ) , because { η ∈ S lh ( t ) | (cid:82) Γ( t ) η = 0 } ⊂ H (Γ( t )) is finite dimensional and thus closed.Note that(5.11) (cid:107)∇ Γ R β v (cid:107) L (Γ( t )) ≤ β max β min (cid:107)∇ Γ v (cid:107) L (Γ( t )) . For fixed t ∈ [0 , T ], the pathwise Ritz projection u p : Ω (cid:55)→ S lh ( t ) of u ∈ L (Ω , H (Γ( t ))) is defined by(5.12) Ω (cid:51) ω → u p ( ω ) = R α ( ω,t ) u ( ω ) ∈ S lh ( t ) . In the following lemma, we state regularity and a -orthogonality. Lemma 5.4.
Let t ∈ [0 , T ] be fixed. Then, the pathwise Ritz projection u p : Ω (cid:55)→ S lh ( t ) of u ∈ L (Ω , H (Γ( t ))) satisfies u p ∈ L (Ω , S lh ( t )) and the Galerkin orthogo-nality (5.13) a ( u − u p , η h ) = 0 ∀ η h ∈ L (Ω , S lh ( t )) . Proof.
By Assumption 2.1 the mappingΩ (cid:51) ω (cid:55)→ α ( ω, t ) ∈ B := { β ∈ L ∞ (Γ( t )) | α min / ≤ β ( x ) ≤ α max } ⊂ L ∞ (Γ( t ))is measurable. Hence by, e.g., [24, Lemma A.5], it is sufficient to prove that themapping B (cid:51) β (cid:55)→ R β ∈ L ( H (Γ( t )) , S lh ( t ))is continuous with respect to the canonical norm in the space L ( H (Γ( t )) , S lh ( t )) oflinear operators from H (Γ( t )) to S lh ( t ). To this end, let β , β (cid:48) ∈ B and v ∈ H (Γ( t ))be arbitrary and we skip the dependence on t from now on. Then, inserting the testfunction ϕ h = ( R β − R β (cid:48) ) v ∈ S lh ( t ) into the definition (5.10), utilizing the stability(5.11), we obtain α min / (cid:107) ( R β (cid:48) − R β ) v (cid:107) H (Γ) ≤ (1 + C P ) (cid:90) Γ β |∇ Γ ( R β (cid:48) − R β ) v | = (1 + C P )( (cid:90) Γ ( β − β (cid:48) ) ∇ Γ R β (cid:48) v ∇ Γ ( R β (cid:48) − R β ) v + (cid:90) Γ β (cid:48) ∇ Γ R β (cid:48) v ∇ Γ ( R β (cid:48) − R β ) v − (cid:90) Γ β ∇ Γ v ∇ Γ ( R β (cid:48) − R β ) v )= (1 + C P ) (cid:18)(cid:90) Γ ( β (cid:48) − β )( ∇ Γ v − ∇ Γ R β (cid:48) v ) ∇ Γ ( R β (cid:48) − R β ) v (cid:19) ≤ (1 + C P ) (cid:107) β (cid:48) − β (cid:107) L ∞ (Γ) (cid:107)∇ Γ ( v − R β (cid:48) v ) (cid:107) L (Γ) (cid:107)∇ Γ ( R β (cid:48) − R β ) v (cid:107) L (Γ) ≤ (cid:18) α max α min (cid:19) (1 + C P ) (cid:107) β (cid:48) − β (cid:107) L ∞ (Γ) (cid:107) v (cid:107) H (Γ) (cid:107) ( R β (cid:48) − R β ) v (cid:107) H (Γ) , where C P denotes the Poincar´e constant in { η ∈ H (Γ) | (cid:82) Γ η = 0 } (see, e.g., [19,Theorem 2.12]). The norm of u p in L (Ω , H (Γ( t ))) is bounded, because Poincar´e’s inequalityand (2.11) lead to α min (cid:90) Ω (cid:107) u p ( ω ) (cid:107) H (Γ( t )) ≤ (1 + C P ) (cid:90) Ω α ( ω, t ) (cid:107)∇ Γ R α ( ω,t ) ( u ( ω )) (cid:107) L (Γ( t )) ≤ (1 + C P ) α max (cid:90) Ω (cid:107)∇ Γ u ( ω ) (cid:107) L (Γ( t )) ≤ (1 + C P ) (cid:107)∇ Γ u (cid:107) L (Ω ,H (Γ( t ))) . This implies u p ∈ L (Ω , S lh ( t )).It is left to show (5.13). For that purpose we select an arbitrary test func-tion ϕ h ( x ) in (5.10), multiply with arbitrary w ∈ L (Ω), utilise w ( ω ) ∇ Γ ϕ h ( x ) = ∇ Γ ( w ( ω ) ϕ h ( x )), and integrate over Ω to obtain (cid:90) Ω (cid:90) Γ( t ) α ( ω, x ) ∇ Γ ( u ( ω, x ) − u p ( ω, x )) ∇ Γ ( ϕ h ( x ) w ( ω )) = 0 . Since { v ( x ) w ( ω ) | v ∈ S lh ( t ) , w ∈ L (Ω) } is a dense subset of V h ( t ), the Galerkinorthogonality (5.13) follows. (cid:3) An error estimate for the pathwise Ritz projection u p defined in (5.12) is estab-lished in the next theorem. Theorem 5.1.
For each fixed t ∈ [0 , T ] , the pathwise Ritz projection u p ∈ L (Ω , S lh ( t )) of u ∈ L (Ω , H (Γ( t ))) satisfies the error estimate (5.14) (cid:107) u − u p (cid:107) H ( t ) + h (cid:107)∇ Γ ( u − u p ) (cid:107) H ( t ) ≤ ch (cid:107) u (cid:107) L (Ω ,H (Γ( t ))) with a constant c depending only on the properties of α as stated in Assumptions 2.1and 2.1 and the shape regularity of Γ h ( t ) .Proof. The Galerkin orthogonality (5.13) and (2.11) provide α min (cid:107)∇ Γ ( u − u p ) (cid:107) H ( t ) ≤ α max inf v ∈ L (Ω ,S lh ( t )) (cid:107)∇ Γ ( u − v ) (cid:107) H ( t ) ≤ α max (cid:107)∇ Γ ( u − I h v ) (cid:107) H ( t ) . Hence, the bound for the gradient follows directly from Lemma 5.1.In order to get the second order bound, we will use a Aubin-Nitsche dualityargument. For every fixed ω ∈ Ω, we consider the path-wise problem to find w ( ω ) ∈ H (Γ( t )) with (cid:82) Γ( t ) w = 0 such that(5.15) (cid:90) Γ( t ) α ∇ Γ w ( ω ) · ∇ Γ ϕ = (cid:90) Γ( t ) ( u − u p ) ϕ ∀ ϕ ∈ H (Γ( t )) . Since Γ( t ) is C , it follows by [19, Theorem 3.3] that w ( ω ) ∈ H (Γ( t )). Insertingthe test function ϕ = w ( ω ) into (5.15) and utilizing the Poincar´e’s inequality, weobtain (cid:107)∇ Γ w ( ω ) (cid:107) L (Γ( t )) ≤ C P α min (cid:107) u − u p (cid:107) L (Γ( t )) . Previous estimate together with the product rule for the divergence imply (cid:107) ∆ Γ w ( ω ) (cid:107) L (Γ( t )) ≤ α min (cid:107) u − u p (cid:107) L (Γ( t )) + C P α (cid:107) α ( ω ) (cid:107) C (Γ( t )) (cid:107) u − u p (cid:107) L (Γ( t )) . Hence, we have the following estimate(5.16) (cid:107) w ( ω ) (cid:107) H (Γ( t )) ≤ C (cid:107) u − u p (cid:107) L (Γ( t )) , SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 17 with a constant C depending only on the properties of α as stated in Assump-tions 2.1 and 2.2. Furthermore, well-known results on random elliptic pdes withuniformly bounded coefficients [7, 9] imply measurablility of w ( ω ), ω ∈ Ω. Inte-grating (5.16) over Ω, we therefore obtain(5.17) (cid:107) w (cid:107) L (Ω ,H (Γ( t ))) ≤ C (cid:107) u − u p (cid:107) H ( t ) . Using again Lemma 5.1, Galerkin orthogonality (5.13), and (5.17), we get (cid:107) u − u p (cid:107) H ( t ) = a ( w, u − u p ) = a ( w − I h w, u − u p ) ≤ α max (cid:107)∇ Γ ( w − I h w ) (cid:107) H ( t ) (cid:107)∇ Γ ( u − u p ) (cid:107) H ( t ) ≤ c (cid:48) h (cid:107) w (cid:107) L (Ω ,H (Γ( t ))) (cid:107) u (cid:107) L (Ω ,H (Γ( t ))) ≤ c (cid:48) ch (cid:107) u − u p (cid:107) H ( t ) (cid:107) u (cid:107) L (Ω ,H (Γ( t ))) . with a constant c (cid:48) depending on the shape regularity of Γ h ( t ). This completes theproof. (cid:3) Remark 5.2.
The first order error bound for (cid:107)∇ Γ ( u − u p ) (cid:107) H ( t ) still holds, if spatialregularity of α as stated in Assumption 2.2 is not satisfied. We conclude with an error estimate for the material derivative of u p that can beproved as in the deterministic setting [17, Theorem 6.2 ]. Theorem 5.2.
For each fixed t ∈ [0 , T ] , the discrete material derivative of thepathwise Ritz projection satisfies the error estimate (cid:107) ∂ • h u − ∂ • h u p (cid:107) H ( t ) + h (cid:107)∇ Γ ( ∂ • h u − ∂ • h u p ) (cid:107) H ( t ) ≤ ch ( (cid:107) u (cid:107) L (Ω ,H (Γ)) + (cid:107) ∂ • u (cid:107) L (Ω ,H (Γ)) )(5.18) with a constant C depending only on the properties of α as stated in Assumptions 2.1and 2.2. Error estimates for the evolving surface finite element discretization.
Now we are in the position to state an error estimate for the evolving surface finiteelement discretization of Problem 2.2 as formulated in Problem 4.1.
Theorem 5.3.
Assume that the solution u of Problem 2.2 has the regularity prop-erties (5.19) sup t ∈ (0 ,T ) (cid:107) u ( t ) (cid:107) L (Ω ,H (Γ( t ))) + (cid:90) T (cid:107) ∂ • u ( t ) (cid:107) L (Ω ,H (Γ( t ))) dt < ∞ and let U h ∈ W h ( V h , H h ) be the solution of the approximating Problem 4.1 with aninitial condition U h (0) = U h, ∈ V h (0) such that (5.20) (cid:107) u (0) − U lh, (cid:107) H (0) ≤ ch holds with a constant c > independent of h . Then the lift u h := U lh satisfies theerror estimate (5.21) sup t ∈ (0 ,T ) (cid:107) u ( t ) − u h ( t ) (cid:107) H ( t ) ≤ Ch with a constant C independent of h . Proof.
Utilizing the preparatory results from the preceding sections, the proof canbe carried out in analogy to the deterministic version stated in [17, Theorem 4.4].The first step is to decompose the error for fixed t into the pathwise Ritz projec-tion error and the deviation of the pathwise Ritz projection u p from the approximatesolution u h according to (cid:107) u ( t ) − u h ( t ) (cid:107) H ( t ) ≤ (cid:107) u ( t ) − u p ( t ) (cid:107) H ( t ) + (cid:107) u p ( t ) − u h ( t ) (cid:107) H ( t ) , t ∈ (0 , T ) . For ease of presentation the dependence on t is often skipped in the sequel.As a consequence of Theorem 5.1 and the regularity assumption (5.19), we havesup t ∈ (0 ,T ) (cid:107) u − u p (cid:107) H ( t ) ≤ ch sup t ∈ (0 ,T ) (cid:107) u (cid:107) L (Ω ,H (Γ( t ))) < ∞ . Hence, it is sufficient to show a corresponding estimate for θ := u p − u h ∈ L (Ω , S lh ) . Here and in the sequel we set ϕ h = φ lh for φ h ∈ L (Ω , S h ).Utilizing (4.7) and the transport formulae (3.6) in Lemma 3.2 and (4.21) inLemma 4.2, respectively, we obtain(5.22) ddt m ( u h , ϕ h ) + a ( u h , ϕ h ) − m ( u h , ∂ • h ϕ h ) = F ( ϕ h ) , ∀ ϕ h ∈ L (Ω , S lh )denoting F ( ϕ h ) := m ( ∂ • h u h , ϕ h ) − m h ( ∂ • h U h , φ h )+ a ( u h , ϕ h ) − a h ( U h , φ h ) + g ( v h ; u h , ϕ h ) − g h ( V h ; U h , φ h ) . (5.23)Exploiting that u solves Problem 2.2 and thus satisfies (2.14) together with theGalerkin orthogonality (5.13) and rearranging terms, we derive(5.24) ddt m ( u p , ϕ h ) + a ( u p , ϕ h ) − m ( u p , ∂ • h ϕ h ) = F ( ϕ h ) , ∀ ϕ h ∈ L (Ω , S lh )denoting(5.25) F ( ϕ h ) := m ( u, ∂ • ϕ h − ∂ • h ϕ h ) + m ( u − u p , ∂ • h ϕ h ) − ddt m ( u − u p , ϕ h ) . We subtract (5.22) from (5.24) to get(5.26) ddt m ( θ, ϕ h ) + a ( θ, ϕ h ) − m ( θ, ∂ • h ϕ h ) = F ( ϕ h ) − F ( ϕ h ) ∀ ϕ h ∈ L (Ω , S lh ) . Inserting the test function ϕ h = θ ∈ L (Ω , S lh ) into (5.26), utilizing the transportLemma 4.2, and integrating in time, we obtain (cid:107) θ ( t ) (cid:107) H ( t ) − (cid:107) θ (0) (cid:107) H (0) + (cid:90) t a ( θ, θ ) + (cid:90) t g (v h ; θ, θ ) = (cid:90) t F ( θ ) − F ( θ ) . Hence, Assumption 2.1 together with (5.9) in Remark 5.1 provides the estimate(5.27) (cid:107) θ ( t ) (cid:107) + α min (cid:90) t (cid:107)∇ Γ θ (cid:107) H ( t ) ≤ (cid:107) θ (0) (cid:107) + c (cid:90) t (cid:107) θ (cid:107) H ( t ) + (cid:90) t | F ( θ ) | + | F ( θ ) | . Lemma 5.2 allows to control the geometric error terms in | F ( θ ) | according to | F ( θ ) | ≤ ch (cid:107) ∂ • h u h (cid:107) H ( t ) (cid:107) θ h (cid:107) H ( t ) + ch (cid:107) u h (cid:107) V ( t ) (cid:107) θ h (cid:107) V ( t ) . SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 19
The transport formula (4.21) provides the identity F ( ϕ h ) = m ( u, ∂ • ϕ h − ∂ • h ϕ h ) − m ( ∂ • h ( u − u p ) , ϕ h ) − g ( v h ; u − u p , ϕ h )from which Lemma 5.3, Theorem 5.2, and Theorem 5.1 imply | F ( θ ) | ≤ ch (cid:107) u (cid:107) H ( t ) (cid:107) θ h (cid:107) V ( t ) + ch ( (cid:107) u (cid:107) L (Ω ,H (Γ( t ))) + (cid:107) ∂ • u (cid:107) L (Ω ,H (Γ( t ))) ) (cid:107) θ h (cid:107) H ( t ) . We insert these estimates into (5.27), rearrange terms, and apply Young’s inequalityto show that for each ε > c ( ε ) such that12 (cid:107) θ ( t ) (cid:107) H ( t ) + ( α min − ε ) (cid:90) t (cid:107)∇ Γ θ (cid:107) H ( t ) ≤ (cid:107) θ (0) (cid:107) H (0) + c ( ε ) (cid:90) t (cid:107) θ (cid:107) H ( t ) + c ( ε ) h (cid:90) t (cid:16) (cid:107) u (cid:107) L (Ω ,H (Γ( t ))) + (cid:107) ∂ • u (cid:107) L (Ω ,H (Γ( t ))) + (cid:107) ∂ • h u (cid:107) H ( t ) + (cid:107) u h (cid:107) V ( t ) (cid:17) . For sufficiently small ε >
0, Gronwall’s lemma implies(5.28) sup t ∈ (0 ,T ) (cid:107) θ ( t ) (cid:107) H ( t ) + (cid:90) T (cid:107)∇ Γ θ (cid:107) H ( t ) ≤ c (cid:107) θ (0) (cid:107) H (0) + ch C h , where C h = (cid:90) T [ (cid:107) u (cid:107) L (Ω ,H (Γ( t )) + (cid:107) ∂ • u (cid:107) L (Ω ,H (Γ( t )) + (cid:107) ∂ • h u (cid:107) H ( t ) + (cid:107) u h (cid:107) V ( t ) ] . Now the consistency assumption (5.20) yields (cid:107) θ (0) (cid:107) H (0) ≤ ch while the stabilityresult (4.22) in Remark 4.1 together with the regularity assumption leads to (5.19) C h ≤ C < ∞ with a constant C independent of h . This completes the proof. (cid:3) Remark 5.3.
Observe that without Assumption 2.2 we still get the H -bound (cid:32)(cid:90) T (cid:107)∇ Γ ( u ( t ) − u h ( t )) (cid:107) H ( t ) (cid:33) / ≤ Ch.
The following error estimate for the expectation E [ u ] = (cid:90) Ω u is an immediate consequence of Theorem 5.3 and the Cauchy-Schwarz inequality. Theorem 5.4.
Under the assumptions and with the notation of Theorem 5.3 wehave the error estimate (5.29) sup t ∈ (0 ,T ) (cid:107) E [ u ( t )] − E [ u h ( t )] (cid:107) L (Γ( t )) ≤ Ch . We close this section with an error estimate for the Monte-Carlo approximationof the expectation E [ u h ]. Note that E [ u h ]( t ) = E [ u h ( t )], because the probabilitymeasure does not depend on time t . For each fixed t ∈ (0 , T ) and some M ∈ N , theMonte-Carlo approximation E M [ u h ]( t ) of E [ u h ]( t ) is defined by(5.30) E M [ u h ( t )] := 1 M M (cid:88) i =1 u ih ( t ) ∈ L (Ω M , L (Γ( t ))) , where u ih are independent identically distributed copies of the random field u h .A proof of the following well-known result can be found, e.g. in [29, Theorem9.22]. Lemma 5.5.
For each fixed t ∈ (0 , T ) , w ∈ L (Ω , L (Γ( t ))) , and any M ∈ N wehave the error estimate (5.31) (cid:107) E [ w ] − E M [ w ] (cid:107) L (Ω M ,L (Γ( t ))) = √ M V ar [ w ] ≤ √ M (cid:107) w (cid:107) L (Ω ,L (Γ( t ))) with V ar [ w ] denoting the variance V ar [ w ] = E [ (cid:107) E [ w ] − w (cid:107) L (Ω , Γ( t )) ] of w . Theorem 5.5.
Under the assumptions and with the notation of Theorem 5.3 wehave the error estimate sup t ∈ (0 ,T ) (cid:107) E [ u ]( t ) − E M [ u h ]( t ) (cid:107) L (Ω M ,L (Γ( t ))) ≤ C (cid:16) h + √ M (cid:17) with a constant C independent of h and M .Proof. Let us first note that(5.32) sup t ∈ (0 ,T ) (cid:107) u h (cid:107) H ( t ) ≤ (1 + C ) sup t ∈ (0 ,T ) (cid:107) u (cid:107) H ( t ) < ∞ follows from the triangle inequality and Theorem 5.3. For arbitrary fixed t ∈ (0 , T )the triangle inequality yields (cid:107) E [ u ]( t ) − E M [ u h ]( t ) (cid:107) L (Ω M ,L (Γ( t ))) ≤(cid:107) E [ u ]( t ) − E [ u h ]( t ) (cid:107) L (Γ( t ))) + (cid:107) E [ u h ( t )] − E M [ u h ( t )] (cid:107) L (Ω M ,L (Γ( t ))) so that the assertion follows from Theorem 5.4, Lemma 5.5, and (5.32). (cid:3) Numerical Experiments
Computational aspects.
In the following numerical computations we con-sider a fully discrete scheme as resulting from an implicit Euler discretization ofthe semi-discrete Problem 4.1. More precisely, we select a time step τ > Kτ = T , set χ kj = χ j ( t k ) , k = 0 , . . . , K, with t k = kτ , and approximate U h ( ω, t k ) by U kh ( ω ) = J (cid:88) j =1 U kj ( ω ) χ kj , k = 0 , . . . , J, with unknown coefficients U kj ( ω ) characterized by the initial condition U h = J (cid:88) j =1 U h, ( X j (0)) χ j and the fully discrete scheme(6.1) 1 τ (cid:0) m kh ( U kh , χ kj ) − m k − h ( U k − h , χ k − j ) (cid:1) + a kh ( U kh , χ kj ) = (cid:90) Ω (cid:90) Γ( t k ) f ( t k ) χ kj for k = 1 , . . . , J . Here, for t = t k the time-dependent bilinear forms m h ( · , · )and a h ( · , · ) are denoted by m kh ( · , · ) and a kh ( · , · ), respectively. The fully discretescheme (6.1) is obtained from an extension of (4.7) to non-vanishing right-handsides f ∈ C ((0 , T ) , H ( t )) by inserting ϕ = χ j , exploiting (4.2), and replacing thetime derivative by the backward difference quotient. As α is defined on the whole SFEM FOR RANDOM ADVECTION-DIFFUSION EQUATIONS 21 ambient space in the subsequent numerical experiments, the inverse lift α − l occur-ring in a h ( · , · ) is replaced by α | Γ h ( t ) , and the integral is computed using a quadratureformula of degree 4.The expectation E [ U kh ] is approximated by the Monte-Carlo method E M [ U kh ] = 1 M M (cid:88) i =1 U kh ( ω i ) , k = 1 , . . . , K, with independent, uniformly distributed samples ω i ∈ Ω. For each sample ω i , theevaluation of U kh ( ω i ) from the initial condition and (6.1) amounts to the solution of J linear systems which is performed by iteratively by a preconditioned conjugategradient method up to the accuracy 10 − .From our theoretical findings stated in Theorem 5.5 and the fully discrete deter-ministic results in [18, Theorem 2.4], we expect that the discretization error(6.2) sup k =0 ,...,K (cid:107) E [ u ]( t k ) − E M [ U kh ] (cid:107) L (Ω M ,L (Γ h ( t k ))) behaves like O (cid:16) h + √ M + τ (cid:17) . This conjecture will be investigated in our numeri-cal experiments. To this end, the integral over Ω M in (6.2) is always approximatedby the average of 20 independent, identically distributed sets of samples.The implementation was carried out in the framework of Dune (DistributedUnified Numerics Environment) [4, 5, 12], and the corresponding code is availableat https://github.com/tranner/dune-mcesfem .6.2.
Moving curve.
We consider the ellipseΓ( t ) = (cid:26) x = ( x , x ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) x a ( t ) + x b ( t ) = 1 (cid:27) , t ∈ [0 , T ] , with oscillating axes a ( t ) = 1 + sin( t ), b ( t ) = 1 + cos( t ), and T = 1. The randomdiffusion coefficient α occurring in a h ( · , · ) is given by α ( x, ω ) = 1 + Y ( ω )4 sin(2 x ) + Y ( ω )4 sin(2 x ) , where Y and Y stand for independent, uniformly distributed random variables onΩ = ( − , f in (6.1) is selected in such a way that for each ω ∈ Ω the exact solution of the resulting path-wise problem is given by u ( x, t, ω ) = sin( t ) (cid:8) cos(3 x ) + cos(3 x ) + Y ( ω ) cos(5 x ) + Y ( ω ) cos(5 x ) (cid:9) , and we set u ( x, ω ) = u ( x, , ω ) = 0.The initial polygonal approximation Γ h, of Γ(0) is depicted in Figure 6.2 for themesh sizes h = h j , j = 0 , . . . ,
4, that are used in our computations. We select thecorresponding time step sizes τ = 1, τ j = τ j − / M = 1, M j = 16 M j − for j = 1 , . . . ,
4. The resulting discretization errors(6.2) are reported in Table 6.2 suggesting the optimal behavior O ( h + M − / + τ ).6.3. Moving surface.
We consider the ellipsoidΓ( t ) = (cid:26) x = ( x , x , x ) ∈ R (cid:12)(cid:12)(cid:12)(cid:12) x a ( t ) + x + x = 1 (cid:27) , t ∈ [0 , T ] , Figure 1.
Polygonal approximation Γ h, of Γ(0) for h = h , . . . , h . h M τ Error eoc( h ) eoc( M ) eoc( τ )1 . . . − . · − . − . . . − . · − . − . . . − . · − . − . . . − . · − . − . . Table 1.
Discretization errors for a moving curve in R .with oscillating x -axis a ( t ) = 1 + sin( t ) and T = 1. The random diffusioncoefficient α occurring in a h ( · , · ) is given by α ( x, ω ) = 1 + x + Y ( ω ) x + Y ( ω ) x , where Y and Y denote independent, uniformly distributed random variables onΩ = ( − , f in (6.1) is chosen such that for each ω ∈ Ω theexact solution of the resulting path-wise problem is given by u ( x, t, ω ) = sin( t ) x x + Y ( ω ) sin(2 t ) x + Y ( ω ) sin(2 t ) x , and we set u ( x, ω ) = u ( x, , ω ) = 0.The initial triangular approximation Γ h, of Γ(0) is depicted in Figure 6.3 for themesh sizes h = h j , j = 0 , . . . ,
3. We select the corresponding time step sizes τ = 1, Figure 2.
Triangular approximation Γ h, of Γ(0) for h = h , . . . , h . h M τ Error eoc( h ) eoc( M ) eoc( τ )1 . . · − — — —0 . − . · − . − . . . − . · − . − . . . − . · − . − . . Table 2.
Discretization errors for a moving surface in R . τ j = τ j − / M = 1, M j = 16 M j − for j = 1 , ,
3. The resulting discretization errors (6.2) are shown in Table 2. Again,
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Institut f¨ur Mathematik, Freie Universit¨at Berlin, 14195 Berlin, Germany
E-mail address : [email protected] Mathematics Institute, University of Warwick, Coventry. CV4 7AL. UK
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Institut f¨ur Mathematik, Freie Universit¨at Berlin, 14195 Berlin, Germany
E-mail address : [email protected] School of Computing, University of Leeds, Leeds. LS2 9JT. UK
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