Accelerated finite elements schemes for parabolic stochastic partial differential equations
- PDF / 626,415 Bytes
- 45 Pages / 439.37 x 666.142 pts Page_size
- 11 Downloads / 207 Views
Accelerated finite elements schemes for parabolic stochastic partial differential equations István Gyöngy1 · Annie Millet2,3 Received: 5 December 2018 / Revised: 20 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract For a class of finite elements approximations for linear stochastic parabolic PDEs it is proved that one can accelerate the rate of convergence by Richardson extrapolation. More precisely, by taking appropriate mixtures of finite elements approximations one can accelerate the convergence to any given speed provided the coefficients, the initial and free data are sufficiently smooth. Keywords Stochastic parabolic equations · Richardson extrapolation · Finite elements Mathematics Subject Classification Primary 60H15 · 65M60; Secondary 65M15 · 65B05
1 Introduction We are interested in finite elements approximations for Cauchy problems for stochastic parabolic PDEs of the form of Eq. (2.1) below. Such kind of equations arise in various fields of sciences and engineering, for example in nonlinear filtering of partially observed diffusion processes. Therefore these equations have been intensively studied in the literature, and theories for their solvability and numerical methods for approximations of their solutions have been developed. Since the computational effort to get
B
Annie Millet [email protected]; [email protected] István Gyöngy [email protected]
1
Maxwell Institute and School of Mathematics, University of Edinburgh, King’s Buildings, Edinburgh EH9 3JZ, UK
2
SAMM (EA 4543), Université Paris 1 Panthéon Sorbonne, 90 Rue de Tolbiac, 75634 Paris Cedex 13, France
3
Laboratoire de Probabilités, Statistique et Modélisation (LPSM, UMR 8001), Paris, France
123
Stoch PDE: Anal Comp
reasonably accurate numerical solutions grow rapidly with the dimension d of the state space, it is important to investigate the possibility of accelerating the convergence of spatial discretisations by Richardson extrapolation. About a century ago Lewis Fry Richardson had the idea in [18] that the speed of convergence of numerical approximations, which depend on some parameter h converging to zero, can be increased if one takes appropriate linear combinations of approximations corresponding to different parameters. This method to accelerate the convergence, called Richardson extrapolation, works when the approximations admit a power series expansion in h at h = 0 with a remainder term, which can be estimated by a higher power of h. In such cases, taking appropriate mixtures of approximations with different parameters, one can eliminate all other terms but the zero order term and the remainder in the expansion. In this way, the order of accuracy of the mixtures is the exponent k + 1 of the power h k+1 , that estimates the remainder. For various numerical methods applied to solving deterministic partial differential equations (PDEs) it has been proved that such expansions exist and that Richardson extrapolations can spectacularly increase the speed of convergence of
Data Loading...
