Random Attractors for the Stochastic Navier–Stokes Equations on the 2D Unit Sphere

In this paper we prove the existence of random attractors for the Navier--Stokes equations on 2 dimensional sphere under random forcing irregular in space and time. We also deduce the existence of an invariant measure

Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties

We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered

Localized linear polynomial operators and quadrature formulas on the sphere

The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere \SS^q ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.Comment: 24 pages 2 figures, accepted for publication in SIAM J. Numer. Ana

Galerkin Approximation for Elliptic PDEs on Spheres

We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the Laplace-Beltrami operator on S n, ω is a non-zero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin metho

A spectral method to the stochastic Stokes equations on the sphere

We construct numerical solutions to the stochastic Stokes equations on the unit sphere with additive noise. By characterising the noise as a tangential vector field, the weak formulation is derived and a spectral method is used to obtain a numerical solution. The theory is illustrated through a numerical experiment. References P. Benner and C. Trautwein. Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations. Technical Report, 2018. URL https://arxiv.org/abs/1809.00911. P. Chen, A. Quarteroni, and G. Rozza. Stochastic optimal Robin boundary control problems of advection-dominated elliptic equations. SIAM J. Numer. Anal., 51(5):2700–2722, 2013. doi:10.1137/120884158. A. Ciraudo, C. D. Negro, A. Herault, and A. Vicari. Advances in modelling methods for lava flow simulation. Commun. SIMAI Cong., 2:1–8, 2007. doi:10.1685/CSC06067. W. Freeden and M. Schreiner. Spherical functions of mathematical geosciences. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer-Verlag, 2009. doi:10.1007/978-3-540-85112-7. M. Ganesh and Q. T. L. Gia. A radial basis Galerkin method for spherical surface Stokes equation. ANZIAM J., 52:C56–C71, 2011. doi:10.21914/anziamj.v52i0.3921. M. Ganesh, Q. T. L. Gia, and I. H. Sloan. A pseudospectral quadrature method for Navier–Stokes equations on rotating spheres. Math. Comput., 80:1397–1430, 2011. doi:10.1090/S0025-5718-2010-02440-8. A. A. Il'in. The Navier–Stokes and Euler equations on two-dimensional manifolds. Math. USSR Sbornik, 69:559–579, 1991. doi:10.1070/sm1991v069n02abeh002116. F. Narcowich, J. Ward, and G. Wright. Divergence-free RBFs on surfaces. J. Fourier Anal. Appl., 13:634–663, 2007. doi:10.1007/s00041-006-6903-2. S. S. Sritharan. Optimal control of viscous flow. SIAM, 1998. doi:10.1137/1.9781611971415. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. Quantum theory of angular momentum. World Scientific, 2008. doi:10.1142/0270