Premium
A posteriori error estimation for elliptic partial differential equations with small uncertainties
Author(s) -
Guignard Diane,
Nobile Fabio,
Picasso Marco
Publication year - 2016
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.21991
Subject(s) - mathematics , a priori and a posteriori , nonlinear system , partial differential equation , finite element method , randomness , discretization , elliptic partial differential equation , mathematical optimization , mathematical analysis , philosophy , statistics , physics , epistemology , quantum mechanics , thermodynamics
In this article, a finite element error analysis is performed on a class of linear and nonlinear elliptic problems with small uncertain input. Using a perturbation approach, the exact (random) solution is expanded up to a certain order with respect to a parameter that controls the amount of randomness in the input and discretized by finite elements. We start by studying a diffusion (linear) model problem with a random coefficient characterized via a finite number of random variables. The main focus of the article is the derivation of a priori and a posteriori error estimates of the error between the exact and approximate solution in various norms, including goal‐oriented error estimation. The analysis is then extended to a class of nonlinear problems. We finally illustrate the theoretical results through numerical examples, along with a comparison with the Stochastic Collocation method in terms of computational costs. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 175–212, 2016