A Fictitious Domain Approach to the Numerical Solution of Pdes In

A ﬁctitious domain approach to the numerical solution of PDEs in stochastic domains

Claudio Canuto1, Tomas Kozubek2 1 Dipartimento di Matematica, Politecnico di Torino, I-10129 Torino, Italy, e-mail: [email protected] 2 Department of Applied Mathematics, VSB-Technical University of Ostrava, 70833 Ostrava, Czech Republik, e-mail: [email protected]

Summary We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.

Key words stochastic partial differential equations – geometric uncertainty – fictitious domain method – finite elements – polynomial chaos expansion – spectral convergence Mathematics Subject Classification (1991): 60H15, 60H35, 65C30, 65N30, 65N35, 65N12

1 Introduction

Stochastic partial differential equations (SPDEs) provide richer mathematical models than standard (deterministic) PDEs, in that they also account for possible uncertainties in the phenomena under modelization. Usually, uncertainties are incorporated into the model by assuming that one or more ingredients which define the PDE problem (the coefficients, the initial or boundary data, the domain) are suitable random variables. The price for dealing with a richer model is obviously an (often significant) increase in the complexity of the problem. This is primarily manifested by the introduction of a (possibly large) number of new independent variables (the stochastic variables), in addition to the standard deterministic variables such as space and time. The numerical solution of a SPDE is therefore an extremely challenging task, which is receiving increasing attention in the literature (see, e.g, the contributions in [26]). The solution of the corresponding deterministic PDE often enters just as a subtask of the whole procedure; the computational effort for solving a single deterministic problem may be amplified, often dramatically, by the number of “degrees of freedom” describing stochasticity. Monte Carlo and quasi Monte Carlo methods ([6]) allow a faithful reproduction of the full statistics of the process, as they mimic in the computer the 2 Claudio Canuto, Tomas Kozubek occurrence of random events in real life; however, the number of solves of the associated deterministic problem may be prohibitively large. On the other hand, often one is just interested in certain statistical quantities of the solution, such as mean value, variance, kurtosis, or probability level sets. Perturbation methods or Neumann-series expansion methods (see, e.g., [19]) yield such information, but require the oscillation of the solution around the mean value to be small. In certain cases, a deterministic PDE satisfied by the statistical quantity of interest can be written and solved numerically (see, e.g., [8,27]). A more general approach, which has received considerable attention in the last decade after the influential work [10], relies upon the global expansion of the random variables defining the PDE into a basis of “elementary” random variables, and the corresponding representation of the solution in terms of such variables (via the Doob-Dynkin lemma). The most popular examples of such expansions are the Karhunen-Loève expansions (see, e.g., [20]), which require the knowledge of the eigenfunctions of the covariance kernel of each random input, and the Polynomial Chaos expansions (also termed (generalized) Wiener Chaos expansions, [30,31]), which exploit the classical technology of weighted orthogonal polynomials. The stochastic problem is then transformed into a deterministic one in higher dimension. The new variables are the images of the basis random variables; they form a coordinate system in a tensor product of intervals of the real line. Discretization in the new variables can be accomplished by a Galerkin projection method ([32]), possibly computationally softened by numerical integration, which may lead to an equivalent collocation method ([23,1]). This paper is focussed on geometric uncertainty, i.e., the main assumption is that the PDE is posed in a domain whose boundary is described by random variables. This situation occurs, e.g., in Aerospace Engineering, where random discrepancies between the mathematical description of aerody- namics bodies used in computer simulation and their actual realization tested in wind tunnels may lead to significant variations in the resulting flow field (see [29,18]). For the sake of simplicity, we con- fine ourselves to the case of a model elliptic PDE. A Polynomial Chaos expansion is used to express the stochastic nature of the parametrization of the boundary of the domain. A natural way to proceed could be to extend the parametrization inside the domain, i.e., to map the random domain onto a fixed one: the PDE of interest would be converted into a new one, posed in a non-random domain but with random coefficients and data (recent results for such equations can be found in [2,1,28]). This strategy is considered and investigated, e.g., in [29,33]. Here, we follow a different approach, based on the Fictitious Domain method ([14,12]). All stochastic domains are assumed to be contained in a fixed, simply-shaped domain, the fictitious domain. The original PDE problem is transformed into a saddle-point problem in the fictitious domain, with the boundary condition on the original stochastic boundary enforced via a Lagrange multiplier. The interest of this approach is that geometric stochasticity is confined at the level of this enforcement. Nonfitted finite element meshes in the fictitious domain and on the stochastic boundary can be used, enhancing flexibility; the stiffness matrix on the fictitious-domain mesh is independent of the geometric stochasticity and can be assembled once and for all. The low accuracy order of the global finite element discretization (due to the singularity created by the Lagrange multiplier) can be compensated for by using a very regular and refined mesh in the fictitious domain. This enables the use of very efficient solvers such as FETI and multigrid type methods, special fast Fourier and cyclic reduction algorithms and special preconditioning techniques. (We refer to [17,16] for applications of the fictitious domain approach to shape optimization and free-boundary problems.) We first give a stochastic version of the fictitious domain formulation (Sects. 2-4), which is next transformed into a deterministic one via the Polynomial Chaos expansion (Sects. 5-6). Discretization is accomplished in Sect. 7 by h-type finite elements in the deterministic variables and orthogonal polynomial projection in the stochastic variables. The discrete variational formulation features tensor-product numerical integration in the latter variables; the resulting scheme is equivalent to a collocation scheme, i.e., to a non-intrusive ([18]) treatment of stochasticity. (The case of a large number of Polynomial A fictitious domain approach to the numerical solution of PDEs in stochastic domains 3

Chaos variables, which may call for “sparse” quadratures, will be considered elsewhere.) We prove the uniform (in all discretization parameters) stability of the numerical method, and the convergence of the approximation, with global and local a-priori error estimates. In particular, we prove that in each subdomain not crossed by any stochastic boundary, the convergence with respect to the Polynomial Chaos truncation is of “spectral” type: the rate of decay of the error is only bounded by the smoothness of the boundary parametrization with respect to the stochastic variables. The eﬃciency of our approach is illustrated in Sect. 8 by two examples.

2 Setting of the problem

Let (Ω, , ) be a complete probability space, where Ω is the set of outcomes, is the σ-algebra of events andF P is the probability measure. For any ω Ω, let D(ω) R2 be a givenF bounded domain, depending onP ω; its boundary Γ (ω) := ∂D(ω) is assumed∈ to be of class⊂ C1,1 or polygonal. We suppose that all domains are contained with their boundaries in a domain Dˆ R2, which will serve as the ﬁctitious domain in the ﬁctitious domain formulation (see Fig. 2.1). ⊂

Dˆ

Γ (ω)

D(ω)

Fig. 2.1. A stochastic domain D(ω) embedded in the ﬁctitious domain Dˆ

For the sake of simplicity, we will be concerned with the following model boundary value problem in D(ω): Find u : D(ω) Ω R such that almost surely (a.s.) in Ω we have × → u( , ω)= f in D(ω), −△ · (ω) ( u( , ω) = 0 on Γ (ω), P · ¡ ¢ where f is a given function in L2(Dˆ). The case of Neumann or mixed boundary conditions ([13,15]) or of random coefficients and data (independent of the random variables describing the domain) ([2, 1]) could be handled at no extra difficulty. So is the extension to a three dimensional random domain. Solving the discrete problem (ω) for any ω Ω using, e.g., the finite element method, means that by varying ω we have to: i) remeshP the new domain∈ D(ω); ii) assemble the new stiffness matrix and the right hand side vector; iii¡ ) solve¢ the new system of linear equations. Thus the efficiency of solving the discrete problems is crucial. Hereafter, we will explore a fictitious domain method with nonfitted meshes as a possible way to increase efficiency: indeed, this approach avoids completely step (i) and partially step (ii), since the stiffness matrix to be “inverted” remains the same for any admissible domain. 4 Claudio Canuto, Tomas Kozubek

3 The ﬁctitious domain (FD) formulation

In this section, we will consider problem (ω∗) for a given outcome ω∗ Ω; we will simplify our notation by setting D := D(ω ), Γ := Γ (ω P) and u = u( , ω ). The weak form∈ of the problem is ∗ ¡∗ ¢ · ∗ Find u H1(D) such that ∈ 0  u vdx = fvdx, v H1(D). P  ∇ · ∇ ∀ ∈ 0 ZD ZD ¡ ¢ Let Dˆ be the ﬁctitious domain containing D and deﬁne the Lagrangian : V M R by L × →

1 2 (v, µ)= v dx fvdx µ, τv Γ , L 2 ˆ |∇ | − ˆ − h i ZD ZD 1/2 1/2 where the symbol ., . denotes the duality pairing between M := H− (Γ ) and H (Γ ), τ : 1 ˆ 1/2 h i 1 ˆ H0 (D) H (Γ ) stands for the trace mapping and V is a closed subspace of H (D). Typical choices → 1 ˆ 1 ˆ 1 ˆ 1 ˆ ˆ ˆ for V are: H (D), H0 (D), or HP (D) = v v H (D), v is periodic on ∂D if D is a cartesian product of intervals. { | ∈ } The reason for introducing the space of the Lagrange multipliers is to fulﬁl the requirement that uˆ solves . Indeed, let us consider the following saddle-point formulation: |D P ¡ ¢ Find (ˆu, λ) V M such that ∈ ×

 uˆ vdx + λ, τv Γ = fvdx, v V, ˆ  Dˆ ∇ · ∇ h i Dˆ ∀ ∈ P  Z Z µ, τuˆ = 0, µ M. ¡ ¢ h iΓ ∀ ∈  The well-posedness of this problem for any f L2(Dˆ) follows from classical results on abstract saddle- point problems, see [4], namely from the fulﬁllment∈ of the following conditions (in fact, (3.3) can be weakened):

c1 = const .> 0: w zdx c1 w V z V , w,z V ; (3.1) ∃ ˆ ∇ · ∇ ≤ k k k k ∀ ∈ ¯ZD ¯ ¯ ¯ c2 = const .> 0: ¯ µ, τz Γ c2 µ¯ M z V , (z, µ) V M; (3.2) ∃ |h¯ i | ≤ k¯ k k k ∀ ∈ × 2 2 α = const .> 0: z dx α z V , z V0; (3.3) ∃ ˆ |∇ | ≥ k k ∀ ∈ ZD µ, τz Γ β = const .> 0 : sup h i β µ M , µ M, (3.4) ∃ z∈V z V ≥ k k ∀ ∈ z6=0 k k where V0 := v V : µ, τv Γ = 0, µ M = v V : v = 0 on ∂D . It is well known that these { ∈ h i ∀ ∈ } { ∈ 1 }ˆ conditions hold true. In particular, (3.3) holds for all z V if V = H0 (D), and holds for all z V0 if 1 ˆ 1 ˆ ∈ ∈ V = H (D) or HP (D), due to the Poincar`einequality. Thus, the following theorem holds.

Theorem 3.1 The saddle-point problem ˆ has a unique solution (û, λ) V M, which satisfies ∂u P ∈ × 3/2 ε uˆ = u and λ = , the jump of the normal derivative of u across Γ . Furthermore, uˆ H − (Dˆ) D ∂n ¡ ¢ ∈ | 2 for any ε> 0 and λ L (Γ ), with uˆ 3 2− + λ 2 C f 2 . £ ∈¤ k kH / ε(Dˆ) k kL (Γ ) ≤ εk kL (Dˆ) A fictitious domain approach to the numerical solution of PDEs in stochastic domains 5

3.1 Discretization of the FD formulation

Problem ˆ will be approximated by using the mixed ﬁnite element method (see [4]). For this purpose P the spaces V and M will be replaced by suitable ﬁnite dimensional subspaces Vh and MH . We now ¡ ¢ 1 ˆ describe a possible construction of Vh and MH . From now on we assume that V = H0 (D). + Let Dˆ be a rectangle and let h , h 0 , be a family of uniform rectangulations of Dˆ. To any such we associate the space {R } → Rh

Vh = vh C(Dˆ) vh Q1(R) R h, vh = 0 on ∂Dˆ ; { ∈ | |R ∈ ∀ ∈ R } Nh thus, Vh contains all continuous piecewise bilinear functions uh = uiϕi over h vanishing on i=1 R ˆ Nh + ∂D, where ϕi i=1 are the Courant basis functions. On the other hand,P let H , H 0 , be a { } NH {R } → regular system of partitions of Γ . More specifically, H := Γr r=1 is a decomposition of Γ into non-overlapping parts whose length does not exceed HR. On any{ } we construct the space RH 2 MH = µH L (Γ ) µH P0(Γr) Γr H , { ∈ | |Γr ∈ ∀ ∈ R } NH NH i.e., MH contains all piecewise constant functions µH = µrχr on H , where χr r=1 are the r=1 R { } characteristic basis functions of Γ . The approximation of Pˆ reads as follows (see [14,12]): r P Find (ûh, λH ) Vh MH such that¡ ¢ ∈ ×  uˆh vh dx + λH vh ds = fvh dx, vh Vh, ˆH  Dˆ ∇ · ∇ Γ Dˆ ∀ ∈ h  Z Z Z P  µ uˆ ds = 0, µ M . ¡ ¢ H h ∀ H ∈ H  ZΓ  To ensure the well-posedness of all ˆH as well as the convergence of the corresponding solutions Ph to the solution of ˆ , we need to satisfy the following Ladyzhenskaja-Babuˇska-Brezzi (LBB)-condition: P ¡ ¢ ¡ ¢ µH vh ds Γ sup Z β µH M , µH MH , (3.5) v V v ≥ k k ∀ ∈ h∈ h k hkV for some β > 0 independent of h and H. It is known that the LBB-condition is satisfied provided the ratio H/h is large enough. This means that the mesh for the Lagrange multipliers has to be coarser than the partition h used for the construction of Vh. See [12] for the case with triangulation, where it is proven that theR condition 3 H/h L, with a fixed positive number L, implies (3.5). Using this property and abstract≤ error≤ estimates (see [4]) we arrive at the following result (see [12]). Theorem 3.2 Let the ratio H/h be sufficiently large and let (û, λ) be the solution to ˆ . Then, there P exists a unique solution (û , λ ) to ˆH and the following estimate holds for any ε> 0: h H Ph ¡ ¢ 1/2 ε uˆ uˆ 1 ¡+ ¢λ λ −1/2 C h f 2 . (3.6) k − hkH (Dˆ) k − H kH (Γ ) ≤ ε − k kL (Dˆ) The rate of convergence is improved in any subdomain compactly contained in D: indeed, if D ′ ⊂ D′ D, one has (see [3]) ⊂ 1 ε uˆ uˆ 1 ′ C h − f 2 . (3.7) k − hkH (D ) ≤ ε k kL (Dˆ) 6 Claudio Canuto, Tomas Kozubek

3.2 Algebraic formulation and its solution

The matrix formulation of ˆH leads to the following system of algebraic equations: Ph ¡ ¢ A BT u f = , P Ã B O !Ã λ ! Ã 0 ! ¡ ¢ where A is the stiﬀness matrix, B is the matrix coupling the primal variable u and the Lagrange multiplier λ, which are the vectors of the nodal values ofu ˆh and of the constant values of λH , respectively, and f is the load vector. The elements of A, f and B are computed as follows:

aij = ϕi ϕj dx, fj = fϕj dx, i,j = 1,...,Nh, (3.8) ˆ ∇ · ∇ ˆ ZD ZD b = ϕ ds, Γ , r = 1,...,N ; j = 1,...,N . (3.9) rj j r ∈ RH H h ZΓr To compute B and f one may use a numerical integration. Note that the information on the geometry of D(ω) is encoded only in B, not in A or f. 1 T To solve P , we use the ﬁrst equation to eliminate the vector u = A− ( B λ+f) from the second one, and we solve the resulting system for λ, − ¡ ¢ 1 T 1 BA− B λ = BA− f, (3.10)

1 T by a conjugate gradient method. The size of BA− B is much smaller than the size of A. The multiplica- 1 tion by A− can be realized, e.g., by Choleski factorization, multigrid approach, domain decomposition method or by using fast solvers based on the Fourier Analysis and the cyclic reduction. In addition, we can use eﬃcient preconditioners to the Schur complement (see, e.g., [14]). Remark 3.1 Let us consider a nonhomogeneous Dirichlet boundary condition u = g on Γ , g H1/2(Γ ). ∈ Then the second set of equations in ˆH has to be replaced by Ph ¡ ¢ µ uˆ ds = µ g ds µ M . (3.11) H h H ∀ H ∈ H ZΓ ZΓ Furthermore, instead of the right hand side (f T , 0T )T in P we have (f T , gT )T , where the elements of g are computed as follows: ¡ ¢ g = g ds, Γ , r = 1,...,N . (3.12) r r ∈ RH H ZΓr

4 The stochastic FD formulation

We go back to the stochastic setting. The FD formulation ˆ suggests the following stochastic FD P formulation: Findu ˆ( , ω) H1(Dˆ) and λ( , ω) M(ω) := H 1/2(Γ (ω)) such that, a.s. in Ω, · ∈ 0 · ∈ ¡ −¢ 1 ˆ uˆ( , ω) vdx + λ( , ω),τv Γ (ω) = fvdx, v H0 (D), Dˆ ∇ · · ∇ h · i Dˆ ∀ ∈ ˆ(ω)  Z Z P  µ, τuˆ( , ω) Γ (ω) = 0, µ M(ω). h · i ∀ ∈ ¡ ¢  A ﬁctitious domain approach to the numerical solution of PDEs in stochastic domains 7

1,1 We assume that, a.s., Γ (ω) is obtained from a reference C or polygonal boundary Γ0 as the image of a piecewise smooth invertible mapping γ0(ω). Precisely, we assume that Γ (ω) = γ0(ω)(Γ0), 1,p where γ0(ω) belongs to C (Γ0), the space of all continuous and piecewise continuously differentiable 2 1 1,p mappings γ : Γ0 R ; its inverse γ0(ω)− exists and belongs to C (Γ (ω)). The function γ0 : Ω C1,p(Γ ) is assumed→ to be a random variable belonging to L (Ω,d ; C1,p(Γ )), i.e., γ is a jointly→ 0 ∞ P 0 0 measurable function on the Borel sets of Γ0 Ω for which there exists a constant g0 > 0 such that 1 × 1 1 γ0(ω) C ,p(Γ0) g0 a.s. in Ω; the same occurs for the inverse mapping, i.e., γ0(ω)− C ,p(Γ (ω)) g0 ka.s. in kΩ. ≤ k k ≤ Let E [X] = X(ω) d (ω) be the expected value of a real-valued random variable X. Let Ω P L2(Ω,d ) = X : Ω R X is a random variable such that E X2 < + be the space of sec- P { R → | ∞} ond order random variables over the probability space (Ω, , ). We denote by L2(Ω,d ; H1(Dˆ)) F P £ ¤ P 0 the space of the random variables v : Ω H1(Dˆ) (i.e., v : D Ω R is jointly measurable and → 0 × → v( , ω) H1(Dˆ) a.s. in Ω) with finite second order moment · ∈ 0 2 2 E v 1 ˆ = E v dx < + . k kH0 (D) ˆ |∇ | ∞ ZD h i £ ¤ 2 1/2 2 1/2 The definition of the space L (Ω,d ; H− (Γ0)) is similar. Finally, the space L (Ω,d ; H− (Γ )) is 2 P 1/2 2 1/2 P defined as follows: µ L (Ω,d ; H− (Γ )) means that µ0 L (Ω,d ; H− (Γ0)), where µ0(ω) 1/2 ∈ P ∈ P1 1/2 ∈ H (Γ ) is defined a.s. in Ω by the conditions µ ,v 0 = µ, v γ− for all v H (Γ ). − 0 h 0 0iΓ h 0 ◦ 0 iΓ (ω) 0 ∈ 0 With such notation at hand, the stochastic FD formulation given at the beginning of the section can be made precise as follows: Findu ˆ L2(Ω,d ; H1(Dˆ)) and λ L2(Ω,d ; H 1/2(Γ )) such that ∈ P 0 ∈ P − 2 1 ˆ E uˆ vdx + E [ λ, τv Γ ]= E fvdx , v L (Ω,d ; H0 (D)), Dˆ ∇ · ∇ h i Dˆ ∀ ∈ P ˆS  ·Z ¸ ·Z ¸ P  E [ µ, τuˆ ] = 0, µ L2(Ω,d ; H 1/2(Γ )). h iΓ ∀ ∈ P − ¡ ¢ Our next step will be to transform this stochastic problem into a purely deterministic one. This will be accomplished by expanding the random variables into polynomial chaos.

5 (Wiener) polynomial chaos

This section is devoted to recalling some basic facts about polynomial chaos (see, e.g., [10,31,24]), as well as to setting the notation. Let Y1(ω),...,Yk(ω),... be a sequence of independent standard Gaussian random variables with zero mean and unit variance, i.e., E [Y ]=0, E [Y Y ]= δ for all k, ℓ 1. On the other hand, given k k ℓ kℓ ≥ the real variable y, let Hn(y) n 0 be the sequence of Hermite polynomials on the real line, satisfying { } ≥

1 y2/2 Hn(y)Hm(y)e− dy = δnm, n,m 0. √2π R ≥ Z N0 N0 Next, denote by y = (yk)k 1 R any infinite sequence of real variables, and by ν = (νk)k 1 N ≥ ∈ ≥ ∈ any infinite sequence of integers which is finite, i.e., such that νk > 0 only for a finite number of indices; let ν = k 1 νk. Define the multidimensional Hermite polynomials of order ν as | | ≥ | |

P ∞ Hν(y)= Hνk (yk); kY=1 8 Claudio Canuto, Tomas Kozubek note that the deﬁnition is meaningful since H0(y) 1, hence, Hν(y) actually depends only on a ﬁnite number of components of y. These polynomials are≡ mutually orthonormal, in the following sense:

∞ 1 2 yk/2 (Hν,Hµ) := Hνk (yk)Hµk (yk)e− dyk = δνµ, ν, µ. √2π R ∀ kY=1 Z Setting Y(ω):=(Yk(ω))k 1 for all ω Ω, the random variables ν : ω Hν(Y(ω)) are independent and with unit variance,≥ since ∈ H 7→

E [ ν µ] = (Hν,Hµ)= δνµ, ν, µ. H H ∀ They form the so-called Wiener chaos (sometimes termed homogeneous chaos or Hermite chaos). The Cameron-Martin theorem [5] states that the family ν so defined forms an orthonormal basis of the space L2(Ω,d ) of the second order random variables{H } over a Gaussian space. The precise result is as follows. P 2 Theorem 5.1 Let Φ L (Ω,d ) and define its coefficients Φν = E [Φ ν] for any finite ν. Then, ∈ P H 2 Φ = Φν ν in L (Ω,d ). H P νXfinite This means, for instance, that we have 2

E Φ Φν ν 0 as N .  − H   → → ∞ ν N | X|≤    N The Cameron-Martin theorem states that Φ(ω) = ϕ(Y(ω)), where ϕ : R 0 R is formally → defined as ϕ(y) = ν finite ΦνHν(y). In many situations of interest, Φ will depend only on a finite number of random variables Yk(ω), say on YK (ω) := (Y1(ω),...,YK (ω)); then, Φ(ω) = ϕ(YK (ω)) K P K with ϕ : R R defined as ϕ(y)= ν NK ΦνHν(y) for y R and satisfying → ∈ ∈ 1P 2 yT y/2 ϕ (y)e− dy < + . (√2π)K RK ∞ Z 2 2 K Thus, for our variable Φ, the condition Φ L (Ω,d ) is equivalent to ϕ L̺(R ), where the weight ∈T P ∈ function ̺ is defined as ̺(y)= 1 e y y/2. The variable y will be termed the stochastic variable, (√2π)K − whereas the spatial variables x and s will be referred to as the deterministic variables. So far, we have focussed on Gaussian random variables. Similar representations can be given for second order random variables over other probabilistic spaces admitting a density function. The system of orthonormal polynomials which gives rise to a generalized polynomial chaos, similar to the Wiener chaos, is determined by the density function; for instance, the uniform density obviously leads to the Legendre polynomials. We refer to [31] for more details. In general terms, a second order random variable Φ depending on a finite number K of mutually independent real random variables Y1(ω),...,YK (ω) with zero mean and unit variance with respect to a density function ρ, can be represented as

Φ(ω)= ϕ(YK (ω)), YK (ω):=(Y1(ω),...,YK (ω)), (5.1) 2 K where ϕ = ϕ(y) satisfies ϕ L̺(I): here, I = I , where I is the interval of the real line on which ρ K ∈ 2 K 2 is defined, and ̺(y)= k=1 ρ(yk). Since L̺(I)= k=1 Lρ(I), a natural orthonormal basis ψν ν NK in this space is provided by the tensor product of a one-dimensional family of orthonormal{ functions} ∈ 2 Q N ψn n N in Lρ(I); we assume that these functions are algebraic polynomials, as it occurs in the most relevant{ } ∈ situations. A fictitious domain approach to the numerical solution of PDEs in stochastic domains 9

6 The deterministic formulation of the stochastic FD problem

We go back to the stochastic formulation ˆS . We assume that the boundary Γ (ω) of D(ω) de- P pends on ω via K mutually independent real random variables Y1(ω),...,YK (ω) with zero mean and unit variance with respect to a density function¡ ¢ ρ defined on some interval I R. Let Y (ω) and ⊆ K ̺ be defined as above. Since we assumed in Sect. 4 that Γ (ω) = γ0(ω)(Γ0), equation (5.1) easily 1,p K yields γ0(ω) = γ0∗(YK (ω)), where γ0∗ = γ0∗(y) is a family of C (Γ0)-mappings defined in I = I , 1 1,p with inverses γ0∗(y)− in C (Γ ∗(y)). Thus, Γ ∗(y) = γ0∗(y)(Γ0) is a parametrization of the set of the admissible boundaries of the stochastic domains D(ω). (A slightly different situation is the one represented in Fig. 2.1, where Γ (ω) = Γ ∗(y) for y = YK (ω) is defined by a suitable interpolation procedure among a set of control nodes, moving along the depicted segments according to the values of the components of y. We refer to Sect. 8, Example 2, for more details.) Sinceu ˆ and λ depend on ω only through Γ (ω), the Doob-Dynkin lemma (see, e.g, [25]) assures that this dependence takes place via YK (ω), i.e., we haveu ˆ( , ω) =u ˆ∗( , YK (ω)) and λ( , ω) = 1 ˆ 1·/2 · · λ∗( , YK (ω)), whereu ˆ∗( , y) H0 (D) and λ∗( , y) H− (Γ ∗(y)), a.e. in I. Conditionu ˆ 2 · 1 ˆ · ∈ 2 · 1 ∈ˆ 2 1/2 ∈ L (Ω,d ; H0 (D)) is then equivalent tou ˆ∗ L̺(I; H0 (D)); similarly, λ L (Ω,d ; H− (Γ )) is P 2 1/2 ∈ ∈ P equivalent to λ∗ L̺(I; H− (Γ ∗)) (with obvious meaning of the notation). We now recall∈ the formula E [Φ]= ϕ(y)̺(y) dy (6.1) ZI which holds for all random variables Φ(ω) = ϕ(Y (ω)) with ϕ L1(I). By applying this formula K ∈ ̺ several times, we transform the stochastic problem ˆS into the following deterministic problem: P Findu ˆ L2(I; H1(Dˆ)) and λ L2(I; H 1/2(Γ )) such that ∗ ∈ ̺ 0 ∗ ∈ ̺ − ∗ ¡ ¢

uˆ∗ v∗ dx ̺(y) dy + λ∗,τv∗ Γ ∗(y)̺(y) dy = fv∗ dx ̺(y) dy, ˆ ∇ · ∇ h i ˆ  ZI ZD ZI ZI ZD 2 1 ˆ D  v∗ L̺(I; H (D)), ˆ  ∀ ∈ 0 P   2 1/2 ¡ ¢ µ∗, τuˆ∗ Γ ∗(y)̺(y) dy = 0, µ∗ L̺(I; H− (Γ ∗)). Ih i ∀ ∈  Z  From now on, we will deal with this problem only. Therefore, we will simplify the notation by dropping the ∗ symbol from all variables. It is understood in the sequel that all unknowns and test functions depend on a deterministic variable (x or s) and on the stochastic variable y I. In order to study the problem, let us introduce the bilinear form (independent of y)∈

1 ˆ a(u,v) := u vdx deﬁned on V V, with V = H0 (D), ˆ ∇ · ∇ × ZD and the companion form

(u,v) := a(u,v) ̺(y) dy deﬁned on , with = L2(I; V ). A V×V V ̺ ZI Conditions (3.1) and (3.3) state that a is continuous and coercive on V , whence is continuous and 1/2 A 2 coercive on , equipped with the norm v = v 1 ̺(y) dy . H (Dˆ) V k kV I k k 0 On the other hand, for any y I, we introduceµZ the bilinear form ¶ ∈ 1/2 b(v, µ; y) := µ, τv deﬁned on V M(y), with M(y)= H− (Γ (y)), h iΓ (y) × 10 Claudio Canuto, Tomas Kozubek and the companion form

(v, µ) := b(v, µ; y) ̺(y) dy defined on , B V×M ZI 1/2 2 2 where = L̺(I; M( )) equipped with the norm µ = µ M(y) ̺(y) dy . Conditions (3.2) M · k kM I k k and (3.4) state that each b is continuous and satisfies an inf-supµZ condition on V ¶ M(y), for constants × c2 = c2(y) and β = β(y) possibly depending on y. However, we need uniformity in y to get similar results for . To this end, we assume that there exist a constant G > 0 such that B 0 1 γ (y) 1,p G , γ (y)− 1,p G , for all y I. (6.2) k 0 kC (Γ0) ≤ 0 k 0 kC (Γ (y)) ≤ 0 ∈ This implies the existence of constants c > 0 and c > 0 independent of y such that for all y I 2 3 ∈ 1 ˆ τv 1/2 c2 v 1 ˆ , v H (D), k kH (Γ (y)) ≤ k kH0 (D) ∀ ∈ 0 1/2 1 ˆ g H (Γ (y)), vg H (D) such that τvg = g and vg 1 ˆ c3 g 1/2 . ∀ ∈ ∃ ∈ 0 k kH0 (D) ≤ k kH (Γ (y)) Thus, the constant β in (3.4) can be chosen independently of y I. ∈ Proposition 6.1 Let assumption (6.2) hold. Then, the form satisfies B

(v, µ) c2 v µ , v , µ , (6.3) |B | ≤ k kV k kM ∀ ∈V ∈M (v, µ) sup B β µ , µ . (6.4) v v ≥ k kM ∀ ∈M ∈V k kV Proof Inequality (6.3) follows immediately from (3.2). The inf-sup condition (6.4) follows from (3.4) by a standard argument which we report for completeness. Let us choose any ε> 0. By (3.4), for any y I and any µ M(y), there exists v(µ) V satisfying v(µ) = 1 and b(v(µ), µ; y) (β ε) µ ∈ . ∈ ∈ k kV ≥ − k kM(y) Given any µ , let us setv ¯(y)= µ(y) M(y)v(µ(y)) a.e. in I. It is easily seen that v¯ = µ ∈M k k k kV k kM and (¯v, µ) (β ε) µ 2 , whence B ≥ − k kM (v, µ) (¯v, µ) sup B B (β ε) µ . v∈V v ≥ v¯ ≥ − k kM v6=0 k kV k kV Letting ε 0+, we get the result. → ⊓⊔ 2 2 ˆ Let (f,v) = (f,v)L2(Dˆ) ̺(y) dy denote the inner product in L̺(I; L (D)). With the notation ZI introduced above, Problem ˆD can be written as follows: Findu ˆ and λ such that P ∈V ∈M ¡ ¢ (ˆu,v)+ (v, λ) = (f,v) v , A B ∀ ∈V ˆD ( (ˆu, µ) = 0, µ . P B ∀ ∈M ¡ ¢

Theorem 6.1 Under the assumptions of Proposition 6.1, Problem ˆD admits a unique solution (ˆu, λ) , satisfying P ∈V×M ¡ ¢ uˆ + λ C f 2 ˆ . k kV k kM ≤ k kL (D) ⊓⊔ A ﬁctitious domain approach to the numerical solution of PDEs in stochastic domains 11

6.1 Smoothness in the stochastic variable

Hereafter we will establish some results about the dependence ofu ˆ on the stochastic variable y. To this end, we introduce a reference domain D0, whose boundary is Γ0, and mappings x = Φ(ξ, y) with ˆ Φ : D0 I D such that Φ( , y)D0 = D(y) and Φ( , y) Γ0 = γ0( , y), for all y I. We assume that × → · · | · ∈ Φ Cr(I; C1(D )) for some r 1, and that Φ( , y) is uniformly invertible in D , i.e., there exists a ∈ 0 ≥ · 0 constant d> 0 such that det JΦ(ξ, y) d for all ξ and y, where JΦ(ξ, y) denotes the Jacobian matrix of Φ( , y) at the point ξ. We also introduce≥ the sets ·

D˜ = D(y) and Γ ∗ = Γ (y). (6.5) y I y I \∈ [∈ 1 Given any neighborhood A of Γ ∗, the mappings Φ( , y) can always be constructed so that Φ( , y)− is independent of y on D˜ A, i.e., Φ(ξ, y)= Φ(ξ) for· all y I and for each ξ such that Φ(ξ) ·D˜ A. We assume that this condition\ is satisﬁed. ∈ ∈ \

r Lemma 6.1 Assume that f H (A), where A is a neighborhood of Γ ∗. Let u0 : D0 I R be defined as u (ξ, y) =u ˆ(Φ(ξ, y∈), y) and let Dα = ∂ α /∂yα, α NK , denote the α-partial× derivative→ 0 y | | ∈ with respect to the y-variable. Then, Dαu ( , y) H1(D ) for all y I and α r, with y 0 · ∈ 0 0 ∈ | | ≤ α D u ( , y) 1 C f 2 + f r . k y 0 · kH0 (D0) ≤ k kL (D(y)) k kH (A) ¡ ¢ The constant C depends on r, Φ r 1 and d, but is independent of y. k kC (I;C (D0)) Proof Sinceu ˆ( , y) satisfies ∆uˆ( , y)= f in D(y) and vanishes on Γ (y), u ( , y) satisfies · − · 0 · T ξ G( , y) ξu0( , y)= f0( , y) in D0, −∇ · ∇ · · (6.6) u ( , y) = 0 on Γ ,  0 · 0 1 T where G(ξ, y) = JΦ(ξ, y)−JΦ(ξ, y)− and f0(ξ, y) = f(Φ(ξ, y)). By the assumptions on Φ, G is uniformly (in ξ and y) positive-definite, hence u ( , y) 1 C f ( , y) 2 C f 2 . k 0 · kH (D0) ≤ 0k 0 · kL (D0) ≤ k kL (D(y)) Let Dy denote any first-order partial derivative in one of the y-variables. By differentiating the previous equation, we deduce that D u ( , y) is the solution of y 0 · T G( , y) ξD u ( , y)= F ( , y) in D , −∇ξ · ∇ y 0 · · 0 D u ( , y) = 0 on Γ ,  y 0 · 0 T  T 2 2 where F (ξ, y)= ξ DyG(ξ, y) ξu0(ξ, y)+(( f)(Φ(ξ, y)) DyΦ(ξ, y). We have DyG ξu0 (L (D0)) , ∇ ∇ ∇ T 2 ∇ 1 ∈ whereas the assumptions on f and Φ yield ( f) DyΦ L (D0). Hence, F H− (D0), so that 1 2 1 ∇ ∈ ∈ Dyu0( , y) H (D0) C f L (D(y)) + f H (A) . k · k ≤ k k k k α We now proceed by induction on the order r′ of differentiation. Any partial derivative Dy u0 of ¡ T¢ α T order α = r vanishes on Γ and satisfies G ξD u = R + S in D , where R is a sum of | | ′ 0 −∇ξ ∇ y 0 ∇ξ 0 products of y-derivatives of G of order r times the gradient of y-derivatives of u of order < r , ≤ ′ 0 ′ whereas S is a sum of products of spatial derivatives of f of order r′ times y-derivatives of Φ of order r . Thus, the right-hand side is again in H 1(D ), and the proof≤ is complete. ≤ ′ − 0 ⊓⊔ 12 Claudio Canuto, Tomas Kozubek

r 1 r 2 Remark 6.1 A similar conclusion holds if f H − (A), provided Φ C (I; C (D0)). Indeed, consid- ∈ ∈ 1 ering for simplicity only the case r = 1, and denoting by , the duality pairing between H− (D0) 1 h · · i and H0 (D0), we have T T 1 T 1 ( f) D Φ, v = ( ξf ) J− D Φ, v = (f , (J− D Φv )) 2 , h ∇ y 0i h ∇ 0 Φ y 0i − 0 ∇ξ Φ y 0 L (D0) T whence ( f) D Φ −1 C Φ 1 2 f 2 . k ∇ y kH (D0) ≤ k kC (I;C (D0))k kL (D(y)) ⊓⊔ Remark 6.2 The previous Lemma concerns the restriction ofu ˆ( , y) to D(y). An analogous result · holds for the restriction ofu ˆ( , y) to Dˆ D(y), after introducing a family of companion mappings · \ Φ ( , y) from some Dˆ D onto Dˆ D(y). ′ · 0 \ 0 \ ⊓⊔ We now draw some consequences from the previous results. The first one concerns the Hölder continuity ofu ˆ with respect to y. Proposition 6.2 Under the assumptions of Lemma 6.1 for r = 1, there exists γ (0, 1] such that ∈ γ uˆ( , y′) uˆ( , y′′) 1 ˆ C f 2 ˆ + f H1(A) y′ y′′ , y′, y′′ I. (6.7) k · − · kH0 (D) ≤ k kL (D) k k k − k ∀ ∈ ³ ´ 1 ˆ Proof Let y′ = y′′ be fixed and let η = y′ y′′ > 0. Sinceu ˆ L∞(I,H0 (D)), it is enough to consider 6 k − k ∈ K the case η η0 for some η0 small enough. Let S be the closed ball in R of radius η containing y′ and y and let≤Γ (S)= Γ (y) : y S . Let Γ (S) denote the closed neighborhood of Γ (S) of radius η; ′′ { ∈ } η our assumptions on γ imply that meas(Γ (S)) cη. Let us split Dˆ as Dˆ = D Γ (S) D , where S 0 η ≤ in ∪ η ∪ out D = D(y) : y S Γ (S) and D = Dˆ (D Γ (S)), and let us estimate the contribution in { ∈ } \ η out \ in ∪ η of each set to the left-hand side of (6.7), starting from Γη(S). T q We observe that there exists 2 < q independent of y such that for any y, uˆ D(y) L (D(y)) q ≤ ∞ ∇ | ∈ and uˆ L (Dˆ D(y)), with uˆ q + uˆ C f 2 , for a constant C ∇ Dˆ D(y) ∈ \ k∇ kL (D(y)) k∇ kLq(Dˆ D(y)) ≤ k kL (Dˆ) | \ 3/2 ε \ independent of y. Indeed, sinceu ˆ H − (Dˆ) for any ε> 0 and the constant Cε in Theorem 3.1 can be taken independent of y, we have∈ q > 4 ε for any ε by the Sobolev imbedding theorem; on the other hand, q can be taken equal to if Γ−(y) is smooth enough. By the Hölder inequality, we have ∞ 2 2 2 uˆ(x, y′) uˆ(x, y′′) dx 2 uˆ(x, y′) dx + 2 uˆ(x, y′′) dx Γη(S) k∇ − ∇ k ≤ Γη(S) k∇ k Γη(S) k∇ k Z Z 2 Z 2 q q q−2 q q 2meas(Γ (S)) q uˆ(x, y′) dx + uˆ(x, y′′) dx , ≤ η  k∇ k k∇ k  ÃZΓη(S) ! ÃZΓη(S) ! q 2   whence, setting γ = 2−q ,

2 2γ uˆ(x, y′) uˆ(x, y′′) dx C f 2 η . k∇ − ∇ k ≤ k kL (Dˆ) ZΓη(S) Let us now focus on D . By definition of Γ (S), it is possible to define the mappings Φ( , y) : in η · D0 D(y) so that they satisfy Φ(ξ, y) = Φ(ξ) for all y S and all ξ D0 such that Φ(ξ) Din; → ∈ ∈ 1 ∈ furthermore, Φ 1 1 C independent of η. Thus, we haveu ˆ(x, y)= u (Φ (x), y) in D for k kC (S;C (D0)) ≤ 0 − in all y S, whence D uˆ(x, y) = D u (Φ 1(x), y) therein, for any first-order partial derivative in one ∈ y y 0 − of the y-variables. Thus, D uˆ 1 C D u 1 C f 2 + f 1 . Writing k y kH (Din) ≤ k y 0kH0 (D0) ≤ k kL (D(y)) k kH (A) 1 ¡ ¢ uˆ(x, y′) uˆ(x, y′′)= y uˆ(x, (1 t)y′ + ty′′) (y′ y′′) dt, ∇ − ∇ ∇ ∇ − · − Z0 A fictitious domain approach to the numerical solution of PDEs in stochastic domains 13 we immediately get

2 2 2 2 2 1 2 1 uˆ(x, y′) uˆ(x, y′′) dx sup yuˆ (H (Din))K y′ y′′ C f L (Dˆ) + f H (A) η . Din k∇ − ∇ k ≤ y S k∇ k k − k ≤ k k k k Z ∈ ³ ´ A similar bound holds in D , whence the result. out ⊓⊔ In general, we cannot expect a higher order of smoothness in y. Indeed, the jump in the spatial normal derivative ofu ˆ on Γ (y) generates a jump in the first-order partial derivatives with respect to the y-variables therein. However, if one “does not cross” Γ (y), thenu ˆ is smooth. Actually, in D˜ (defined in (6.5)), we can differentiate the equation ∆u( , y)= f infinitely many times in y, showing α − · that all partial derivatives Dy u of any order are harmonic, hence smooth, therein. A more precise result is as follows. α 1 ˜ Proposition 6.3 Under the assumptions of Lemma 6.1, one has Dy u H (D A) for all y I and α r, with ∈ \ ∈ | | ≤ α Dy u H1(D˜ A) C f L2(Dˆ) + f Hr(A) . k k \ ≤ k k k k ³ ´ α α 1 Proof The assumption on Φ stated before the lemma guarantees that Dy u(x, y)= Dy u0(Φ− (x), y) for all x D˜ A, y I. ∈ \ ∈ ⊓⊔ Remark 6.3 We refer to [28] for results about the analyticity (in y) of the solution of an elliptic PDE with coefficients and data depending analytically on y; such a situation occurs, e.g., if a (truncated) Karhunen-Loève expansion of these quantities is used. We conclude with a smoothness result for the Lagrange multiplier λ. Proposition 6.4 Under the assumptions of Lemma 6.1, one has Dαλ H 1/2(Γ ) for all y I and y 0 ∈ − 0 ∈ α α r 1, with D λ −1/2 C f 2 + f r . In particular, if r 2, λ is Lipschitz | | ≤ − k y 0kH (Γ0) ≤ k kL (Dˆ) k kH (A) ≥ 0 1/2 continuous with respect to y, with values³ in H− (Γ0). ´

Proof We recall that λ = ∂uˆ and we observe that ∂uˆ = ∂u0 , the conormal derivative of ∂n Γ (y) ∂n D(y) ∂nG | 0 u with respect to the matrix G (see (6.6)). The result³ is an easy´ consequence of Lemma 6.1. 0 £ ¤ ⊓⊔

7 Discretization of the deterministic formulation

In this section, we discretize Problem ˆD with respect to both the deterministic variables x and s, and the stochastic variable y. To thisP end, from now on we assume that conditions (6.2) on the ¡ ¢ 1/2 1/2 family of mappings γ0 hold true. Given any µ H− (Γ (y)), let µ0 H− (Γ0) be defined by the condition µ ,v = µ, v γ (y) 1 for all∈ v H1/2(Γ ); thanks∈ to (6.2), there exist constants h 0 0iΓ0 h 0 ◦ 0 − iΓ (y) 0 ∈ 0 c4,c4′ > 0 such that 1/2 c µ −1/2 µ −1/2 c′ µ −1/2 , µ H− (Γ (y)), a. e. in I. (7.1) 4k kH (Γ (y)) ≤ k 0kH (Γ0) ≤ 4k kH (Γ (y)) ∀ ∈ 1/2 Let us choose a finite dimensional subspace M0,H of M0 = H− (Γ0), made of piecewise constant functions on Γ as described in Sect. 3.1. Then, for all y I, we define the finite dimensional subspace 0 ∈ MH (y) of M(y) as M (y)= µ M(y) : µ M , H { ∈ 0 ∈ 0,H } which is again a space of piecewise constant functions on Γ (y). 14 Claudio Canuto, Tomas Kozubek

K We now turn to the polynomial chaos and apply a truncation. Let ψν(y)= k=1 ψνk (yk) denote the K multivariate orthonormal polynomial of multi-degree ν = (ν1, . . . , νK ) N for the density function ∈ Q K ̺(y). Let us set, as usual, ν = max νk : 1 k K and ν 1 = ν = k=1 νk. Given any k k∞ { ≤ ≤ } k k | | cut-off integer N > 0, let QN (I) = span ψν : ν N be the space of the polynomials of degree { k k∞ ≤ } P N in each variable, and PN (I) = span ψν : ν 1 N be the space of the polynomials of total degree≤ N. Obviously, P (I) = Q (I){ if K k=k 1;≤ otherwise} Q (I) P (I) Q (I), with ≤ N N [N/K] ⊂ N ⊂ N (N+K)! K dim PN (I)= N!K! whereas dim QN (I) = (N + 1) . Truncation to PN (I) is the most appropriate in stochastic problems, as it yields the approximation of precisely all moments of order N of a random ≤ variable. On the other hand, truncation to QN (I) is typical of spectral methods for PDEs in tensor- product domains (see, e.g., [7]); it has the advantage of allowing the use of simple Gaussian quadrature rules of high precision, whose knots are also optimal polynomial interpolation points. We assume in the sequel that the number K of independent random variables which define the domain uncertainty is small; consequently, we choose QN (I) as the approximation space in the stochastic variables. As we will see, such a choice allows the splitting of the resulting discrete problem into a set of independent fictitious domain problems, a feature that - for moderate values of K - by far compensates the larger dimension of QN (I) with respect to PN (I). (Comments on the alternative use of PN (I) are contained in Remark 7.1 below.) Obviously, when K is large enough, truncation to PN (I) becomes mandatory, and the tensor-product quadrature formula (7.3) should be more efficiently replaced by a Smolyak formula constructed on a sparse grid in I (see, e.g., [11]) and exact for all polynomials in P2N (I). We refer to [24] for more details (see also [28] for related ideas). The use of sparse approximations in the present framework will be investigated elsewhere. As mentioned above, we introduce a tensor-product high-precision quadrature formula in I. To this end, let (yq, ρq) : 0 q N be the set of the nodes and weights of the Gauss quadrature formula for the measure{ ρ in I≤, for≤ which} we have

N p(y) ρ(y) dy = p(y )ρ , p P (I). (7.2) q q ∀ ∈ 2N+1 I q=0 Z X (Truncated formulas with enhanced properties could be used in the Hermite or Laguerre case [22].) By tensorization we get the formula

p(y) ̺(y) dy = p(yq)̺q, p Q (I), (7.3) ∀ ∈ 2N+1 I q Q Z X∈ N K K where QN = q = (q1,...,qK ) N : q N , yq = (yq1 ,...,yqK ) and ̺q = k=1 ρqk . If Z is a { ∈ k k∞ ≤ } 2 Banach space and N = Z QN (I) is the subspace of = L̺(I; Z) of the Z-valued polynomials of degree N in eachZ of the y⊗-variables, we have, as a consequenceZ of (7.3), Q ≤ 1/2 2 z 2 = z( , yq) ̺q , z N . (7.4) k kL̺(I;Z)  k · kZ  ∀ ∈ Z q Q X∈ N   We now introduce the finite dimensional space = V Q (I) (7.5) Vh,N h ⊗ N ⊂V of the functions v(x, y) = ν ∞ N vν(x)ψν(y) such that vν Vh. Similarly, we define the finite dimensional subspace of k k ≤ ∈ MP = µ : µ M Q (I) ; (7.6) MH,N { ∈M 0 ∈ 0,H ⊗ N } A fictitious domain approach to the numerical solution of PDEs in stochastic domains 15

thus, µ0 has the form µ0(s, y)= ν ∞ N µ0,ν(s)ψν(y) with µ0,ν M0,H ; note that µ( , y) MH (y) k k ≤ ∈ · ∈ for all y I. Finally, we introduce an approximation N of the bilinear form , obtained by replacing exact integration∈ over I with numericalP quadrature; precisely,B we set B

: R, (v, µ)= b(v( , yq), µ( , yq); yq) ̺q. (7.7) BN Vh,N ×MH,N → BN · · q Q X∈ N

We are ready to consider the following finite dimensional approximation of Problem ˆD : Find uˆ and λ such that P h,N ∈Vh,N H,N ∈MH,N ¡ ¢ (û ,v )+ (v , λ ) = (f,v ), v , A h,N h,N BN h,N H,N h,N ∀ h,N ∈Vh,N ˆD h,H,N ( N (ûh,N , µH,N ) = 0, µH,N H,N . P B ∀ ∈M ¡ ¢ Problem ˆD can be given an equivalent formulation of collocation type (see [23,1] for other Ph,H,N examples of collocation schemes for SPDEs). Indeed, thanks to (7.3), the bilinear form (u,v) coincides ¡ ¢ on = V Q (I) with the discrete form A VN ⊗ N

(u,v)= a(u( ., yq),v( ., yq))̺q. AN q Q X∈ N

For any q Q , let Lq(y) denote the Lagrange basis function in Q (I) associated with the node yq, ∈ N N i.e., Lq(yq′ )= δqq′ for all q Q . Thanks to (7.3), we have ′ ∈ N

Lp(y)Lr(y)̺(y) dy = Lp(yq)Lr(yq)̺q = δpr̺p, p, r Q . ∀ ∈ N I q Q Z X∈ N

Choosing vh,N (x, y)= vh(x)Lq(y) h,N and µH,N H,N such that (µH,N )0(s, y) = (µH )0(s)Lq(y) ∈V ∈M D and cancelling the weight ̺q from both sides of each equation in ˆ , we immediately see that, Ph,H,N for all q Q ,u ˆ q =u ˆ ( , yq) V and λ q = λ ( , yq) M (yq) are the solution of the ∈ N h, h,N · ∈ h H, H,N · ¡∈ H ¢ saddle point problem in V M (yq): h × H

a(ˆuh,q,vh)+ b(vh, λH,q; yq) = (f,vh), vh Vh, ∀ ∈ (7.8) ( b(ˆuh,q, µH ; yq) = 0, µH MH (yq). ∀ ∈

Conversely, given the solutions (ˆu q, λ q) V M (yq) of these problems for all q Q , the h, H, ∈ h × H ∈ N solution (ˆu , λ ) of problem ˆD can be expressed as h,N H,N ∈Vh,N ×MH,N Ph,H,N ¡ ¢ uˆh,N (x, y)= uˆh,q(x)Lq(y), (λH,N )0(s, y)= (λH,q)0(s)Lq(y). (7.9) q Q q Q X∈ N X∈ N

In practice, one solves the (N +1)K independent ﬁctitious domain problems (7.8), then one recovers uˆh,N and λH,N via (7.9). Hence, the treatment of the stochastic variable is non-intrusive (see, e.g., [23,18]), i.e., it does not require any modiﬁcation of the deterministic code used for solving (7.8). Each independent problem has the algebraic structure P discussed in Sect. 3.2; considerable savings come from the fact that the information on the geometry of D(y) is encoded only in B but not in A and f. ¡ ¢ 16 Claudio Canuto, Tomas Kozubek

7.1 Well-posedness and stability of the discrete problem

The bilinear form restricted to is obviously uniformly continuous and coercive. In order to A Vh,N study the form N , we ﬁrst observe that conditions (6.2) and (3.5) easily imply the following property: if the ratio H/hBis large enough, there exists a constant β > 0 such that

µH vh ds Γ (y) sup Z β µH M(y), µH MH (y), y I. (7.10) v V v ≥ k k ∀ ∈ ∀ ∈ h∈ h k hkV Proposition 7.1 Assume that the ratio H/h is large enough, so that (7.10) holds. Then, there exist constants c˜2, β˜ > 0 independent of h, H, N such that

N (vh,N , µH,N ) c˜2 vh,N µH,N , vh,N h,N , µH,N H,N , (7.11) |B | ≤ k kV k kM ∀ ∈V ∈M N (vh,N , µH,N ) sup B β˜ µH,N , µH,N H,N . (7.12) M vh,N h,N vh,N ≥ k k ∀ ∈M ∈V k kV Proof Thanks to (6.2) and (3.2), we have

(v , µ ) c v ( , yq) µ ( , yq) ̺q |BN h,N H,N | ≤ 2 k h,N · kV k H,N · kM(yq) q Q X∈ N 1/2 1/2 2 2 c v ( , yq) ̺q µ ( , yq) ̺q . ≤ 2  k h,N · kV   k H,N · kM(yq)  q Q q Q X∈ N X∈ N     1/2 2 Identity (7.4) applied to vh,N yields q Q vh,N ( , yq) V ̺q = vh,N . On the other hand, ∈ N k · k k kV by (7.1) and again (7.4) applied to (µ³H,NP )0, we get ´ 1/2 1/2 2 1 2 µ ( , yq) ̺q (µ ) ( , yq) ̺q  k H,N · kM(yq)  ≤ c  k H,N 0 · kM0  q Q 4 q Q X∈ N X∈ N   1  1/2 = (µ ) ( , y) 2 ̺(y) dy c k H,N 0 · kM0 4 µZI ¶ 1/2 c4′ 2 c4′ µH,N ( , y) M(y) ̺(y) dy = µH,N . ≤ c k · k c k kM 4 µZI ¶ 4 This yields (7.11) withc ˜ = (c /c )c . Let us now consider (7.12). Given µ , set µq = 2 4′ 4 2 H,N ∈ MH,N µH,N ( , yq). For any q QN , by (7.10) there existsv ¯q Vh, such that v¯q V = 1 and b(¯vq, µq; yq) · ∈ ∈ k k 2 ≥ β µq y . Set v∗ = µq y v¯q, so that we have b(v∗, µq; yq) β µq and v∗ V = k kM( q) q k kM( q) q ≥ k kM(yq) k qk µq M(yq). Let vh,N∗ be the unique function in h,N which interpolates the values vq∗ at the quadrature k k V 2 nodes q QN . Then, N (vh,N∗ , µH,N ) β q Q µq M(y ) ̺q and ∈ B ≥ ∈ N k k q P 1/2 1/2 2 2 vh,N∗ = vq∗ V ̺q = µq M(y )̺q . k kV  k k   k k q  q Q q Q X∈ N X∈ N     c4 ˜ Since, as above, the last term is ′ µH,N , we get (7.12) with β = (c4/c4′ )β. ≥ c4 k kM ⊓⊔ A ﬁctitious domain approach to the numerical solution of PDEs in stochastic domains 17

As a consequence of the proposition, we have the well-posedness and the stability of Problem ˆD . Ph,H,N ¡Theorem¢ 7.1 Under the assumptions of Proposition 7.1, Problem ˆD admits a unique solution Ph,H,N (ˆu , λ ) , satisfying h,N H,N ∈Vh,N ×MH,N ¡ ¢

uˆh,N + λH,N C f 2 ˆ k kV k kM ≤ k kL (D) for a constant C independent of h, H, N.

7.2 Convergence analysis

We now provide error bounds between the solutions of the exact problem ˆD and the discrete P problem ˆD . For the sake of definiteness, hereafter we assume that the density function is the h,H,N ¡ ¢ Gaussian one;P results similar to those established below hold for the most common probability densities as well. We¡ first consider¢ the Gauss-Hermite interpolation error with respect to the stochastic variable y I. ∈ Lemma 7.1 Let Z be a Banach space and let z C0(I; Z) be a Z-valued continuous function. Let ∈ I z = Z Q (I) be the interpolant of z at the nodes yq, q Q . If z is a Hölder-continuous N ∈ ZN ⊗ N ∈ N function of exponent γ (0, 1], i.e., if there exists a constant R > 0 such that z(y′) z(y′′) Z R y y γ for all y , y∈ I, then, k − k ≤ k ′ − ′′k ′ ′′ ∈ γ/2 z IN z 2 CγRN − . k − kL̺(I;Z) ≤ If, in addition, z satisfies Dαz C for all y I and all α = r, with 1 r N + 1, then k y kZ ≤ ∈ | | ≤ ≤ r/2 z IN z 2 CrN − z r,∞ I , k − kL̺(I;Z) ≤ | |W ( ;Z) α where z W r,∞(I;Z) = max α =r Dy z L∞(I;Z). | | | | k k Proof ([21]) We proceed by induction on K, assuming at first that K = 1. Since I is exact on , N ZN we have z IN z = (z pN ) IN (z pN ) for all pN N . By (7.2) and the smoothness of Hermite’s weight, we− get − − − ∈ Z

N N 2 2 2 1/2 ρq IN (z pN ) 2 = (z pN )(yq) Z ρq = (z pN )(yq) Z ρ(yq) k − kL̺(I;Z) k − k k − k ρ(y )1/2 q=0 q=0 q X X 1/4 2 1/2 C (z p )ρ ∞ ρ(y) dy. ≤ k − N kL (I;Z) ZI 2 1/4 2 1/2 On the other hand, one also has z pN 2 (z pN )ρ ∞ ρ(y) dy. Introducing the k − kL̺(I;Z) ≤ k − kL (I;Z) I weighted best-approximation error in ZN R 1/4 1/4 2 EN (z; ρ , ) = inf (z pN )ρ L∞(I;Z), ∞ pN N k − k ∈Z we thus have 1/4 z IN z 2 CEN (z; ρ , ). k − kL̺(I;Z) ≤ ∞ 18 Claudio Canuto, Tomas Kozubek

Classical results on weighted best-approximation (see, e.g., [9]) yield

1/√N ωr(z,t; ρ1/4, ) E (z; ρ1/4, ) C ∞ dt, 1 r N + 1, N ∞ ≤ t ≤ ≤ Z0 where ωr(z,t; φ, p) is the weighted modulus of smoothness r r ω (z,t; φ, p) = sup ∆hφ( )z( ) Lp(I;Z), 0 1 and the result hold up to K 1. Let us split K 1 K − y = (y′, yK ) with y′ I′ = I − and yK I, and let IN′ and IN be the interpolation operators in I′ and ∈ K ∈ K I, respectively. Since IN = IN′ IN , we have, with obvious notation, z IN z = z IN′ z +IN′ (z IN z), whence ⊗ − − − 1/2 1/2 2 K 2 z I z 2 (z I z)( , y ) 2 ′ ρ(y ) dy + (z I z)(y ′ , ) 2 ̺ ′ N L̺(I;Z) N′ K L (I ;Z) K K N q′ L (I;Z) q′ k − k ≤ k − · k ̺′  k − · k ρ  µZI ¶ q′ Q′ X∈ N   We conclude by using the one-dimensional result to bound the first integral on the right-hand side and the recurrence assumption plus (7.3) to bound the second integral on the right-hand side. ⊓⊔ We apply the previous lemma to the functionu ˆ : I H1(Dˆ), whose Hölder continuity is guaranteed → 0 by Proposition 6.2, and to its restriction to D˜ A, whose smoothness in y is assured by Proposition \ 1/2 6.3. We also apply it to the function λ0 : I H− (Γ0), whose smoothness in y is guaranteed by Proposition 6.4; we denote by I λ the→ function defined by the condition (I λ) = I (λ ). N ∈M N 0 N 0 Proposition 7.2 Let the assumptions of Lemma 6.1 hold with r = 1, and let γ (0, 1] satisfy (6.7). Then, ∈ γ/2 uˆ IN uˆ CγN − f 2 ˆ + f H1(A) . k − kV ≤ k kL (D) k k Furthermore, for any r 1 for which the assumptions³ of Lemma 6.1 are´ satisfied, one has ≥ r/2 uˆ IN uˆ L2(I;H1(D˜ A)) CrN − f L2(Dˆ) + f Hr(A) k − k ̺ \ ≤ k k k k and, if r 2, ³ ´ ≥ (r 1)/2 λ IN λ CrN − − f 2 ˆ + f Hr(A) . k − kM ≤ k kL (D) k k ⊓⊔ ³ ´ ˆD ˆD We are ready for estimating the error between the solutions of problems and h,H,N . Observe thatu ˆ and λ depend continuously on y by Propositions 6.2 and 6.4; consequently,P P for all ¡ ¢ ¡ ¢ yq Q ,u ˆq =u ˆ( , yq) V and λq = λ( , yq) M(yq) are the solution of the saddle point problem ∈ N · ∈ · ∈ in V M(yq): × a(ûq,v)+ b(v, λq; yq) = (f,v), v V, ∀ ∈ (7.13) ( b(ûq, µ; yq) = 0, µ M(yq). ∀ ∈ A fictitious domain approach to the numerical solution of PDEs in stochastic domains 19

Since (7.8) is the discretization of such a saddle point problem, Theorem 3.2 applies; we thus have, for all yq Q and all ε> 0, ∈ N 1/2 ε uˆq uˆ q + λq λ q C h − f 2 , k − h, kV k − H, kM(yq) ≤ ε k kL (Dˆ) as well as 1 ε uˆq uˆh,q H1(D˜ A) Cεh − f L2(Dˆ). k − k \ ≤ k k The constants Cε can be chosen independent of yq. By (7.4), these inequalities imply the estimates 1/2 ε IN uˆ uˆh,N + IN λ λH,N Cεh − f 2 ˆ , k − kV k − kM ≤ k kL (D) and 1 ε IN uˆ uˆh,N L2(I;H1(D˜ A)) Cεh − f L2(Dˆ). k − k ̺ \ ≤ k k The triangle inequality and Proposition 7.2 yield the ﬁnal result, expressed by the following theorem. Theorem 7.2 Let uˆ and λ be the solution of Problem ˆD . Let the h,N ∈ Vh,N H,N ∈ MH,N Ph,H,N assumptions of Lemma 6.1 hold with r = 1, and let γ (0, 1] satisfy (6.7); let the ratio H/h be ¡ ¢ suﬃciently large. Then, ∈

γ/2 1/2 ε uˆ uˆh,N CγN − + Cεh − f 2 ˆ + f H1(A) . (7.14) k − kV ≤ k kL (D) k k For any r 1 for which the assumptions³ of Lemma 6.1´ are ³ satisﬁed, one has ´ ≥ r/2 1 ε uˆ uˆh,N L2(I;H1(D˜ A)) CrN − + Cεh − f L2(Dˆ) + f Hr(A) (7.15) k − k ̺ \ ≤ k k k k and, if r 2, ³ ´ ³ ´ ≥ (r 1)/2 1/2 ε λ λH,N CrN − − + Cεh − f 2 ˆ + f Hr(A) . k − kM ≤ k kL (D) k k ⊓⊔ ³ ´ ³ ´ This result proves the convergence of each moment ofu ˆh,N to the corresponding moment ofu ˆ, as N and h 0. If r can be chosen arbitrarily large, the rate of convergence is faster than algebraic in N→at ∞ all points→ x which have a positive distance from the boundary of all domains D(y).

Remark 7.1 Let us briefly comment on using PN (I), rather than QN (I), for the polynomial chaos truncation. By this we mean that we replace QN (I) by PN (I) in the definitions (7.5) and (7.6) of the finite dimensional spaces and ; all other ingredients of the discretization scheme ˆD Vh,N MH,N Ph,H,N remain unchanged. Numerical evidence indicates that the inf-sup condition (7.12) holds as well for ¡ ¢ the new spaces. This implies a convergence theorem analogous to Thm. 7.2. As far as the algebraic formulation is concerned, let us set N = (N+K)! 1 and let us label ψ Npc the chosen orthonormal pc N!K! − { k}k=0 basis in P (I). Problem ˆD is then equivalently written as N Ph,H,N ¡ ¢ A˜ B˜T u˜ ˜f = , P˜ Ã B˜ O˜ !Ã λ˜ ! Ã 0˜ ! ¡ ¢ where A O O B B B ··· 00 01 ··· 0Npc O A O B B B    10 11 1Npc  A˜ = ··· , B˜ = ··· ,      ············   ············   O O A   B B B     Npc0 Npc1 NpcNpc   ···   ···  20 Claudio Canuto, Tomas Kozubek

T ˜ T ˜ T u˜ = (u0, u1,..., uNpc ) , λ = (λ0, λ1,..., λNpc ) and f = (f0, f1,..., fNpc ) . The entries aij of A and fj of f are computed by (3.8). On the other hand, computing the entries of B˜ is diﬀerent due to the stochastic dependence. Precisely, we have for 0 k,l N ≤ ≤ pc [B ] = ϕ ds ψ (y )ψ (y )̺ , r = 1,...,N , j = 1,...,N . kl rj j k q l q q H h (7.16) q QN Z X∈ hΓr(yq) i System P˜ can be solved, e.g., by Uzawa’s iterations. Note that the multiplication by B˜, B˜T and the application of A˜ 1 can be done in parallel due to simple regular block structure. However, the ¡ ¢ − “intrusive” character of the method is apparent from the non-diagonal structure of the matrix B˜.

8 Numerical examples

In this section, we illustrate the eﬃciency of our approach on two model examples with homogeneous and nonhomogeneous Dirichlet boundary conditions, respectively. Example 1 is chosen in such way that we know the exact solution. We exhibit the expected exponential convergence of our method with respect to the polynomial chaos order. In Example 2, no analytic solution is available. Therefore basic Monte Carlo (MC) simulation without using any special optimization technique is used to validate the result.

Example 8.1 Let Dˆ := (0, 1) (0, 1) be the ﬁctitious domain. Let us consider a single stochastic variable ×

1 Dˆ 20

15 D(y) y 10 a b

5 Γ (y)

0 0.2 0.25 0.3 0.35 0.4 0 1 Fig. 8.1. The density function ρ(y) Fig. 8.2. The domain D(y) y = y, associated with a normal distribution Y N[y, σ] with y = (a+b)/2, σ = (b a)/8; the density function ρ(y) is truncated from R to the interval∼ I = [a, b] with a = 0.2, b = 0.4− (see Figure 8.1). Despite this variable has not zero mean and unit variance, a simple change of variable would allow us to satisfy the assumptions posed in the previous sections. The variable y represents the radius of the circular domain D(y) Dˆ: ⊂ D(y) := x = (x ,x ) R2 (x 0.5)2 + (x 0.5)2 < y2 , 1 2 ∈ | 1 − 2 − where x is the deterministic© variable (see Figure 8.2). We will be concerned withª the deterministic problem A ﬁctitious domain approach to the numerical solution of PDEs in stochastic domains 21

u(x, y)=4 in D(y), −△ (y) ( u(x, y) = 0 on Γ (y), P ¡ ¢ whose exact solution is u(x, y)= y2 (x 0.5)2 (x 0.5)2. (8.1) − 1 − − 2 − In order to make all our experiments independent of the auxiliary boundary condition chosen on ∂Dˆ, the exact solutionu ˆ(x, y) in Dˆ will not be taken as the solution of Problem ˆ , but it will be re- defined as the function u extendend by 0 outside D(y). The same post-processingP will be applied to ˆD ¡ ¢ the discrete solutionu ˆh,N (x, y) of Problem h,H,N , at the nodes of the finite-element mesh. We monitor the convergence behavior ofP our method with respect to the polynomial chaos order ¡ ¢ 2 N and the mesh-size h. We consider the mean u and the variance σu of the computed solution 2 uˆh,N , as well as their exact values uex and σuex . Means and variances obtained with h = 1/128, H/h = 3, N = 0,..., 6, are shown in Figures 8.3 and 8.4, respectively. Graphs are restricted to the 1 line L = (x1, 2 ), x1 [0, 1] ; the dash-dotted vertical lines indicate the intervals of variation of the boundary{ of the domain.∈ } The corresponding relative errors in L2(Dˆ) and L (Dˆ), defined for p =2 or p = as ∞ ∞ 2 2 u uex Lp(D˜) σu σuex Lp(D˜) δmean = k − k , δvar = k − k , Lp(D˜) u Lp(D˜) σ2 k exkLp(D˜) k uex kLp(D˜) are reported in Figure 8.5 as functions of N, again for fixed h = 1/128, H/h = 3. The slow convergence, due to the singularity on Γ (y) and predicted by estimate (7.14), is clearly documented. On the other hand, if we restrict the error to the intersection D˜ of all the stochastic domains (see (6.5)), we obtain spectral convergence, as predicted by estimate (7.15) and clearly documented in Figure 8.6. Note that in D˜ the exact solution (8.1) is indeed a quadratic polynomial in y; hence, the errors in Figure 8.6 for N 2 are due to the spatial discretization only. As a comparison, we report in Figure 8.7 the relative≥ errors on mean and variance obtained by a standard direct Monte-Carlo method, as a function of the number of trails Nmc; in this case, the convergence is not monotonic and very slow. Finally, the errors obtained by our method by varying h = 1/64, 1/128, 1/256, 1/512 and keeping N = 6, H/h = 3 2 fixed are documented in Figure 8.8. Since Nh = h− , we observe quadratic convergence in h in the 2 L - and L∞-norms, a result coherent with the interior estimate (3.7), which measures the error in the H1-norm. A deeper insight on the polynomial chaos behavior is obtained by monitoring the Hermite coefficients of the exact and approximate solutions. Fig. 8.9 displays the discrete Hermite coefficients of the N exact solutionu ˆ, defined for ν = 0,...,N asu ˆν(x) = q=0 uˆ(x, yq)Hν(yq)̺q, as well as those of the N approximate solutionu ˆh,N , defined as (ûh,N )ν(x) = Pq=0 uˆh,N (x, yq)Hν(yq)̺q. As above, we choose N = 6, h = 1/128 and H/h = 3; graphs are restricted to the line L = (x , 1 ), x [0, 1] . Both the P { 1 2 1 ∈ } fast decay in D˜ and the slow decay in Γ ∗ (see (6.5)) are clearly documented. For all modes, except perhaps for the highest one, there is a good agreement between the exact and the approximate values. The last set of plots stresses the fact that the slow decay of the Hermite coefficients is only due to the singularity across Γ (y) created by the extension procedure of the solution. We monitor the exact and approximate solutions along a family of curves x = xλ(y) which stay on the same side of Γ (y) for all y, although they are not confined in D˜. The precise definition is x (y) = (0.5+ λd(y), 0.5), where y λ λ is a parameter satisfying λ y and d(y) = . The discrete Hermite coefficients of | | ≤ y(1 + (y y)2) − uˆ(xλ(y), y) andu ˆh,N (xλ(y), y) are plotted in Fig. 8.10 versus the point 0.5+ λ; as above, we choose N = 6, h = 1/128 and H/h = 3. As expected, a fast decay of the coefficients ofu ˆ(x(y), y) is observed, as a manifestation of the smoothness of this function of the variable y. The coefficients ofu ˆh,N (xλ(y), y) 22 Claudio Canuto, Tomas Kozubek

−5 x 10 0.1 Exact 12 Exact N = 0 N = 0 0.08 N = 1 10 N = 1 N = 2 N = 2 N = 3 8 0.06 N = 3 N = 4 N = 4 N = 5 6 N = 5 0.04 N = 6 N = 6 4

0.02 2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fig. 8.3. Mean solution Fig. 8.4. Variance −2 10 −2 10 2 Mean in L ∞ −4 Mean in L 10 −4 2 10 Var. in L ∞ Var. in L −6 −6 10 10 Mean in L2 ∞ −8 −8 10 Mean in L 10 Var. in L2 ∞ Var. in L −10 −10 10 10 0 2 4 6 0 2 4 6 N N Fig. 8.5. Relative errors (PC) in Dˆ, h ﬁxed Fig. 8.6. Relative errors (PC) in D˜, h ﬁxed −2 10

−6 10 −4 10

−8 −6 10 10 Mean in L2 Mean in L2 ∞ −10 −8 Mean in L 10 ∞ 10 2 Mean in L Var. in L 2 ∞ Var. in L Var. in L ∞ −10 −12 Var. in L 10 10 0 1 N 2 3 3 4 N 5 6 10 10 mc 10 10 10 10 h 10 10 Fig. 8.7. Relative errors (MC), h ﬁxed Fig. 8.8. Relative errors (PC), N ﬁxed

decay at a slower rate around the points 0.5 y, since the approximate solutionu ˆh,N is globally aﬀected by the singularity along Γ (y); note, however,± that the higher-order coeﬃcients are almost two order of magnitude smaller than the corresponding ones in Fig. 8.9.

Example 8.2 Let Dˆ := (0, 1) (0, 1) be again the fictitious domain. Let y = (y , y ) be the vector × 1 2 stochastic variable, associated with two independent normal distributions Yk N[y, σ], k = 1, 2, defined as in the previous example with a = 0.25 and b = 0.35. In a polar coordinate∼ system centered A fictitious domain approach to the numerical solution of PDEs in stochastic domains 23

Exact 0.08 Computed

0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 Mode 0

−3 −3 x 10 x 10 3.5 1.5 Exact Exact 0.01 Exact Computed Computed 3 1 Computed 0.008 2.5 0.5 0.006 2 0 1.5 0.004 −0.5 1 0.002 0.5 −1

0 0 −1.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 1 Mode 2 Mode 3

−3 −3 −3 x 10 x 10 x 10 1.5 1.5 1.5 Exact Exact Exact 1 Computed 1 Computed 1 Computed

0.5 0.5 0.5

0 0 0

−0.5 −0.5 −0.5

−1 −1 −1

−1.5 −1.5 −1.5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 4 Mode 5 Mode 6 Fig. 8.9. Hermite coeﬃcients - ﬁxed point case

at x0 = (0.5, 0.5), consider the control points Ck, k = 0,..., 15, whose angles are ϕk = kπ/8 and whose radii are constant, rk = 0.3, except for k = 5 and k = 6: for these control points, the radii are given by the variables y2 and y1, respectively (see Figure 8.11). The boundary Γ (y) is obtained by connecting the control points via a piecewise Bèzier curve of the second order, identified by the Bèzier triples (Mk, Ck+1,Mk+1), with Mk = (Ck + Ck+1)/2 for k = 0,..., 15 and C16 = C0, M16 = M0. All possible configurations of the stochastic domain D(y) are obtained by moving the control nodes C5 and C6 along the depicted lines. We consider the problem

u(x, y) = 60 in D(y), −△ (y) ( u(x, y)= g on Γ (y), P ¡ ¢ 24 Claudio Canuto, Tomas Kozubek

Exact Computed 0.08

0.06

0.04

0.02

0 0 0.2 0.4 0.6 0.8 1 Mode 0

−3 −4 −5 x 10 x 10 x 10 10 Exact Exact Exact 3.5 Computed Computed 3 Computed 8 3 2 2.5 6 1 2 4 1.5 0

1 −1 2 0.5 −2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 1 Mode 2 Mode 3

−5 −5 −5 x 10 x 10 x 10 5 5 5 Exact Exact Exact Computed Computed Computed

0 0 0

−5 −5 −5 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 4 Mode 5 Mode 6 Fig. 8.10. Hermite coefficients - moving point case where 0, ϕ [ π, 0], g(ϕ)= ∈ − 1 cos(2ϕ), ϕ (0, π). ( − ∈ 1 The mean and variance along the line L = (x1, 2 ), x1 [0, 1] of the computed solutionsu ˆh,N , with h = 1/128, H/h = 3, are reported in Figs.{ 8.12 and∈ 8.13, respectively,} for several values of N. The comparison with an “exact” solution, obtained with a very fine discretization in both h and N, documents the accuracy of the discretization method. Figs. 8.14 and 8.15 provide comparisons between the results produced by basic Monte Carlo (MC) simulation, for different numbers of trials Nmc, and second order Polynomial Chaos (PC) results, obtained by solving 9 independent deterministic problems (N = 2). While the Monte Carlo approxi- A fictitious domain approach to the numerical solution of PDEs in stochastic domains 25

−5 1 x 10 ˆ D 5 Exact Exact N = 0 b N = 0 y1 N = 1 1.5 N = 1 4 a N = 2 N = 2 N = 3 N = 3 D(y) 3 N = 4 1 N = 4 2

Γ (y) 0.5 1

0 0 0 0 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fig. 8.11. Geometry of D(y) Fig. 8.12. Mean solution Fig. 8.13. Variance mation of the mean value is good already for moderate numbers of trials, an acceptable approximation of the variance is obtained only with a number of trials in the order of several hundreds. Finally, Fig. 8.16 shows the discrete Hermite coeﬃcients of the “exact” and approximate solutions for N = 3 (and h = 1/128, H/h = 3) along the line L. The high accuracy of the approximation, predicted by the theory since the exact solution is smooth in y on L, is clearly documented.

−5 x 10 PC, 2. order PC, 2. order 1.8 5 MC, N = 10 MC, N = 10 1.6 MC, N = 20 MC, N = 20 MC, N = 50 4 MC, N = 50 1.4 MC, N = 100 MC, N = 100 MC, N = 200 MC, N = 200 1.2 MC, N = 500 3 MC, N = 500 1 MC, N = 1000 MC, N = 1000 0.8 2 0.6

0.4 1 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Fig. 8.14. Polynomial Chaos vs Monte Carlo, Fig. 8.15. Polynomial Chaos vs Monte h ﬁxed: mean solution Carlo, h ﬁxed: variance

Acknowledgements Illuminating discussions with Giovanni Monegato and Giovanni Pistone are gratefully acknowledged. This research was supported by the European project “Breaking Complexity”, n. HPRN-CT-2002-00286, by the Italian grant MIUR-PRIN 2004 prot. 2003011441-004 and by the grants IAA1075402 and 1ET400300415 of the Grant Agency of the Czech Academy of Sciences. 26 Claudio Canuto, Tomas Kozubek

Exact Computed 1.5

0.5

0 0 0.2 0.4 0.6 0.8 1 Mode 0,0

−3 −3 x 10 −5 x 10 7 x 10 Exact Exact 0 6 Computed Exact 2 Computed Computed −2 5 −4 1.5 4 −6 3 1 −8 2 0.5 −10 1 −12 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 1,0 Mode 0,1 Mode 2,0

−5 −4 −6 x 10 x 10 x 10 Exact 0 Exact 5 Exact 6 Computed Computed Computed −1 4 0 −2 2 −5 −3 0

−2 −4 −10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 1,1 Mode 0,2 Mode 3,0 −6 x 10 −6 −5 x 10 x 10 Exact 2 Exact Exact Computed 2.5 2 Computed Computed 1 2 1.5 0 1.5 1 −1 1 0.5 −2 0.5 0 −3 0 −0.5 −4 −0.5 −1 −5 −1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Mode 2,1 Mode 1,2 Mode 0,3 Fig. 8.16. Hermite coeﬃcients of Example 2 A ﬁctitious domain approach to the numerical solution of PDEs in stochastic domains 27

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