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SIAM J. on Numerical Analysis / Volume 50 / Issue 1

SIAM J. Numer. Anal. 50, pp. 216-246 (31 pages)

Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients

Julia Charrier

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We consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous Dirichlet boundary conditions. We focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness. This model is frequently used in hydrogeology. We approximate this coefficient by a finite dimensional noise using a truncated Karhunen–Loève expansion. We give estimates of the corresponding error on the solution, both a strong error estimate and a weak error estimate, that is, an estimate of the error commited on the law of the solution. We obtain a weak rate of convergence which is twice the strong one. In addition, we give a complete error estimate for the stochastic collocation method in this case, where neither coercivity nor boundedness is stochastically uniform. To conclude, we apply these results of strong and weak convergence to two classical cases of covariance kernel choices, the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel, yielding explicit weak and strong convergence rates.

© 2012 Society for Industrial and Applied Mathematics

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KEYWORDS

AMS Subject Headings

65N15, 65C20, 60H35, 76S05

PUBLICATION DATA

ISSN

0036-1429 (print)  
1095-7170 (online)

Publisher

$abstract.society.value is a Member of CrossRef Society for Industrial and Applied Mathematics

ARTICLE DATA

History
Received June 29, 2010
Accepted October 31, 2011
Published online February 09, 2012
Digital Object Identifier
http://dx.doi.org/10.1137/100800531

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