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Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients

Julia Charrier 1
1 SAGE - Simulations and Algorithms on Grids for Environment
Inria Rennes – Bretagne Atlantique , IRISA-D1 - SYSTÈMES LARGE ÉCHELLE
Abstract : 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, we have then to 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 then estimates of the corresponding error on the solution, both a strong error estimate and a weak error estimate, that is to say an estimate of the error commited on the law of the solution. We obtain a weak rate of convergence wich is twice the strong one. Besides this, we give a complete error estimate for the stochastic collocation method in this case, where neither coercivity nor boundedness are 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.
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Submitted on : Tuesday, April 5, 2011 - 2:03:32 PM
Last modification on : Thursday, January 20, 2022 - 4:20:36 PM
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  • HAL Id : inria-00490045, version 3


Julia Charrier. Strong and weak error estimates for the solutions of elliptic partial differential equations with random coefficients. [Research Report] RR-7300, INRIA. 2010. ⟨inria-00490045v3⟩



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