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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 signif- icant) 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 corre- sponding 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-Lo`eve 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 stochas- tic 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 in- terest 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 polyno- mial 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 smooth- ness 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.
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