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hp ‐FEM for second moments of elliptic PDEs with stochastic data. II: Exponential convergence for stationary singular covariance functions
Author(s) -
Pentenrieder Bastian,
Schwab Christoph
Publication year - 2012
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.20690
Subject(s) - mathematics , exponential function , mathematical analysis , finite element method , elliptic partial differential equation , covariance , gaussian , partial differential equation , elliptic curve , diagonal , elliptic operator , convergence (economics) , moment (physics) , geometry , physics , statistics , quantum mechanics , economics , thermodynamics , economic growth , classical mechanics
Abstract We prove exponential rates of convergence of a class of hp Galerkin Finite Element approximations of solutions to a model tensor nonhypoelliptic equation in the unit square □ = (0, 1) 2 which exhibit singularities on ∂□ and on the diagonal Δ = {( x, y ) ∈ □ : x = y }, but are otherwise analytic in □. As we explained in the first part (Pentenrieder and Schwab, Research Report, Seminar for Applied Mathematics, 2010) of this work, such problems arise as deterministic second moment equations of linear, second order elliptic operator equations Au = f with Gaussian random field data f . © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012