Numerical analysis of partial differential equations with random coefficients, applications to hydrogeology
type: Dataset
description: This work presents some results about probabilistic and deterministic numerical methods for partial differential equations with stochastic coefficients, with applications to hydrogeology. We first consider the steady flow equation in porous media with a homogeneous lognormal permeability coefficient, including the case of a low regularity covariance function. We establish error estimates, both in strong and weak senses, of the error in the solution resulting from the truncature of the Karhunen-Loève expansion of the coefficient. Then we establish finite element error estimates, from which we deduce an extension of the existing error estimate for the stochastic collocation method along with an error estimate for a multilevel Monte-Carlo method. We finally consider the coupling of the previous flow equation with an advection-diffusion equation, in the case when the uncertainty is important and the correlation length is small. We propose the numerical analysis of a numerical method, which aims at computing the mean velocity of the expansion of a pollutant. The method consists in a Monte-Carlo method, combining a finite element method for the flow equation and an Euler scheme for the stochastic differential equation associated to the advection-diffusion equation, seen as a Fokker-Planck equation.
identifier: NNT: 2011DENS0030
issued: 2026-05-15T22:10:28.318098
language: fr
modified: 2026-05-15T22:10:28.318104
contact point: Organization
  fn: CCSD
keyword: coefficient-lognormal
keyword: developpement-de-karhunen-loeve
keyword: euler-scheme-for-stochastic-differential-equations
keyword: finite-element-method
keyword: infoeu-reposemanticsdoctoralthesis
keyword: karhunen-loeve-expansion
keyword: lognormal-coefficient
keyword: mathmath-gmmathematics-mathgeneral-mathematics-mathgm
keyword: methode-de-collocation-stochastique
keyword: methode-de-monte-carlo
keyword: methode-de-monte-carlo-multi-niveaux
keyword: methode-delements-finis
keyword: monte-carlo-method
keyword: multilevel-monte-carlo-method
keyword: quantification-des-incertitudes
keyword: schema-deuler-pour-des-equations-differentielles-stochastiques
keyword: stochastic-collocation-method
keyword: theses
keyword: uncertainty-quantification
landing page: https://theses.hal.science/tel-00625092

Numerical analysis of partial differential equations with random coefficients, applications to hydrogeology
type: Distribution
format: HTML
issued: 2026-05-15T22:10:28.404525
modified: 2026-05-15T22:10:28.277545
access URL: https://theses.hal.science/tel-00625092

test_moissonnage_selune
type: Agent