Variational methods for PDEs aplied to stochastic partial differential equations

Våge, Gjermund

doi:10.7146/math.scand.a-13828

Cited by 35 publications

(32 citation statements)

References 6 publications

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“…To provide conditions on f such that g → f ♦g is a continuous linear operator in S −1,k,0 (D), we introduce the Banach space F l (D) [35]. Given l ∈ R, we define the Banach space [35], [32] …”

Section: Preliminaries On White Noise Analysis and Stochastic Sobolevmentioning

confidence: 99%

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Mixed finite element method for the heat diffusion equation in a random medium

Paquet

Korikache

2010

Stochastics

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We introduce the dual mixed method for the heat evolution equation in a polygonal domain D with a random diffusion coefficient K and heat flux − → p = K♦ − → ∇u, ♦ denoting the Wick product. We prove a-priori error estimates for the semi-discrete solution ( − → p h , u h ) of lowest order of the dual mixed method having K-dimensional polynomial chaos expansion of degree N . Due to the reentrant corner of the polygonal domain D, appropriate refinement rules must be imposed on the family of triangulations in order to recapture convergence of order one in space.

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Section: Preliminaries On White Noise Analysis and Stochastic Sobolevmentioning

confidence: 99%

“…F l (D) is a commutative Banach algebra for the Wick product ( [35], prop.6, p.123) with 1 as unity. Moreover, if f ∈ F l (D) and g ∈ S −1,k,0 (D) with k ≤ 2l, then the Wick product is a well defined element of…”

Section: Preliminaries On White Noise Analysis and Stochastic Sobolevmentioning

confidence: 99%

Mixed finite element method for the heat diffusion equation in a random medium

Paquet

Korikache

2010

Stochastics

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“…See also [37], which emphasises SFEM for reliability analysis. See [40,38] for an analysis of SFEM discretizations based on white noise analysis .…”

Section: The Stochastic Finite Element Methodmentioning

confidence: 99%

Computational aspects of the stochastic finite element method

Eiermann

Ernst

Ullmann

2007

Comput. Visual Sci.

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“…The white noise approach exploits the built-in set of stochastic spaces, such as Hida or Kondratiev spaces [11,12], or even larger exponential spaces [16]. The traditional approach [17,20,21], etc., has to select a stochastic space and then to study the largest possible class of equations admitting a solution in that space. The difference of our approach is that we select the stochastic space that is in some sense optimal for the particular equation under consideration.…”

Section: Assumption a The Expectation Of The Highest Order (Differenmentioning

confidence: 99%

Elliptic equations of higher stochastic order

Lototsky¹,

Rozovskiĭ²,

Wan³

2010

ESAIM: M2AN

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Abstract. This paper discusses analytical and numerical issues related to elliptic equations with random coefficients which are generally nonlinear functions of white noise. Singularity issues are avoided by using the Itô-Skorohod calculus to interpret the interactions between the coefficients and the solution. The solution is constructed by means of the Wiener Chaos (Cameron-Martin) expansions. The existence and uniqueness of the solutions are established under rather weak assumptions, the main of which requires only that the expectation of the highest order (differential) operator is a non-degenerate elliptic operator. The deterministic coefficients of the Wiener Chaos expansion of the solution solve a lower-triangular system of linear elliptic equations (the propagator). This structure of the propagator insures linear complexity of the related numerical algorithms. Using the lower triangular structure and linearity of the propagator, the rate of convergence is derived for a spectral/hp finite element approximation. The results of related numerical experiments are presented.Mathematics Subject Classification. 35R60, 65L60, 60H15, 60H35.

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Variational methods for PDEs aplied to stochastic partial differential equations

Cited by 35 publications

References 6 publications

Mixed finite element method for the heat diffusion equation in a random medium

Mixed finite element method for the heat diffusion equation in a random medium

Computational aspects of the stochastic finite element method

Elliptic equations of higher stochastic order

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