具有随机强迫的二维稳态Navier-Stokes系统的侵入和非侵入多项式混沌近似

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arXiv:2107.11720v1 [math.NA] 25 Jul 2021

INTRUSIVE AND NON-INTRUSIVE POLYNOMIAL CHAOS

APPROXIMATIONS FOR A TWO-DIMENSIONAL STEADY STATE

NAVIER-STOKES SYSTEM WI T H RANDOM FORCING

S. V. LOTOTSKY, R. MIKULEVICIUS, AND B. L. ROZOVSKY

Abstract. While convergence of polynomial chaos approximation for linear equa-

tions is relatively well understood, a lot less is known for non-linear equations.

The paper investigates this convergence for a particular equa tion with quadratic

nonlinearity.

July 27, 2021

1. Introduction

There are two main ways to study an equation with random input. One way is to use

deterministic tools for each particular realization of randomness; in what follows, we

call it path-wise approach. An alternative, which we call random field approach,

is to consider the random input as an additional independent variable in the equation,

along with space and/or time.

Questions such as existence/uniqueness/regularity of the solution are often addressed

with a combination of the two approaches; cf. [18, 19] for ordinary diﬀerential equa-

tions and [17, 24, 25] for equations with partial derivatives.

The diﬀerence between t he two approaches becomes noticeable in numerical compu-

tations; see, for example, [13, 35]. Path-wise approach leads to repeated numerical

solutions of the underlying equation for various realizations of the random input; a

typical example is Monte Carlo simulations. In computational terms, this approach

is non-intrusive, because no new numerical procedures are required to solve the

equation compared to the deterministic case.

The random ﬁeld approach reduces the problem to a ﬁxed system of deterministic

equations via a stochastic Galerkin approximation; in many cases, the result is a

generalized polynomial chaos (gPC) expansion [35, Chapter 6]. In computational

terms, this appro ach is intrusive, because the resulting system is more complicated

than the original equation and requires diﬀerent numerical procedures to obtain a

solution.

2020 Mathematics Subject Classiﬁcation. 35Q30, 35R60, 60H35, 65N15 , 65N30, 65N35, 76D05,

76M22.

Key words and phrases. Gauss Quadrature, Generalized Polynomial Chaos, Stochastic Galerkin

Approximation.

SVL: Research suppor ted by ARO Grants W911N-16-1-0103 and FA9550-21-1-0015.

RM and BLR: Research supported by ARO Grant W911N-16-1-0103.

2 S. V. LOTOTSKY, R. MIKULEVICIUS, AND B. L. ROZOVSKY

The stochastic collocation method [35, Chapter 7], with sampling at pre-determined

realizations of the rando m input, somewhat bridges the gap between pure r andom

sampling (Monte Carlo) and complete elimination of randomness (g PC). In com-

putational terms, the method is non-intrusive [3, 31]. In this paper, we consider

the disc rete projection, o r pseudo-spectral version of the method, when t he sampled

solution is used to approximate the coeﬃcients in the chaos expansion via Gauss

quadrature.

For many, although apparently not all [8], equations, various empirical studies

[15, 28, 32, etc.] suggest that the stochastic Galerkin approximation method, with

a ﬁxed computational cost, can be a much more eﬃcient way to study statistical

properties of the solution tha n Monte Carlo or stochastic collocation methods. In the

case of nonlinear equations, this exp erimental success has yet to be fully justiﬁed the-

oretically; for linear equations, the picture is rather clear; see, for example, [9, 20, 21]

as well as [22, Chapter 5] a nd [30, Section 8.3].

Accordingly, our objective in this paper is to carry out a comparative theoretical

analysis of an intrusive and a non-intrusive approximations for a particular nonlinear

equation. Speciﬁcally, we consider the stationary Navier-Stokes system in a smooth

bounded planar domain with zero boundary conditions and with randomness in the

external force, and we establish a priori error bounds for both approximations.

The paper is org anized as follows. Section 2 describes the model and introduces the

necessary function spaces. Section 3 introduces the stochastic Galerkin approximation

and gives the proof of convergence. Section 4 investigates a non-intrusive pseudo-

spectral approximation. Section 5 puts the results in a broader context.

Throughout the paper, G is a bounded domain in R

with area |G| and suﬃciently

regular (e.g. locally Lipschtiz) boundary ∂G. We use the following convention with

the notations of various function spaces and their elements: if X denotes a space of

scalar ﬁelds f on G, then X denotes the corresponding space of vector ﬁelds f, and

X denotes the collection of X-valued random elements f .

2. The Setting

Let (Ω, F, P) be a probability space such that the L

(Ω) is a separable Hilbert space

and has a complete orthogonal basis {P

, n ≥ 0}: with

c(n) = EP

, (2.1)

every ζ ∈ L

(Ω) can be represented as

ζ =

k≥0



ζP

)

c(k)

In what follows, we always assume that the basis {P

, n ≥ 0} has the following

property: for every m, n ≥ 0, there a r e ﬁnitely many real numb ers A

m,n;l

, l ≥ 0, such

that

l≥0

m,n;l

; (2.2)

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