We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity, and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.
International audienceWe 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-Lo'eve 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
We study here explicit flux-splitting finite volume discretizations of multi-dimensional nonlinear scalar conservation laws perturbed by a multiplicative noise with a given initial data in L 2 (R d ). Under a stability condition on the time step, we prove the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation. Keywords : Stochastic PDE • first-order hyperbolic equation • Itô integral • multiplicative noise • finite volume method • flux-splitting scheme • Engquist-Osher scheme • Lax-Friedrichs scheme • upwind scheme • Young measures • Kruzhkov smooth entropy. Mathematics Subject Classification (2000) : 35L60 • 60H15 • 65M08 • 65M12
We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in $L^{2}(\R^d)$. After establishing the well-posedness theory for solutions of such kind of stochastic problems, we prove under a stability condition on the time step the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation
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