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A virtual element method for stochastic Stokes equations. (English) Zbl 1443.65358
Summary: In this paper, a virtual element method for the stochastic Stokes equations driven by an additive white noise is proposed and analyzed. The velocity is approximated by the lowest-order virtual element which is originally designed for the Poisson equation and the projection is also taken as the one originally for the Poisson equation, while the pressure is approximated by the traditional discontinuous piecewise constant element. For stable approximations, we adopt a stabilization associating with the pressure jumps. We show the inf-sup condition and derive the stability. We moreover obtain the error estimates in various norms and the estimates of the expectation of the errors through the Green function. Numerical results on polygonal mesh are presented to illustrate the performance of the proposed method and the theoretical results obtained.
MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
76M35 Stochastic analysis applied to problems in fluid mechanics
35R60 PDEs with randomness, stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
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