A posteriori error estimation for partial differential equations with small uncertainties
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Cita com:
hdl:2117/333997
Tipus de documentText en actes de congrés
Data publicació2015
EditorCIMNE
Condicions d'accésAccés obert
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Abstract
Abstract. Partial differential equations (PDEs) are widely used for modelling problems
in many fields such as physics, biology or engineering. In general, any problem is affected by a
certain level of uncertainty, due to the intrinsic variability of the system or the
inability to adequately characterize all the involved input. Nowadays, it is common to
include uncertainty in the mathematical models describing such complex systems, using for instance
probability theory characterizing the uncertain input by random variables or more generally by
random fields.
The purpose of this work is to perform a priori and a posteriori error analysis for
PDEs with random input data. We consider small uncertainties and adopt a perturbation approach
expanding the exact (random) solution u of a given problem up to a certain order as
u(x, ω) = u0(x)+ εu1(x, ω)+ ε2u2(x, ω)+ ... (1)
where ε is a parameter that controls the amount of randomness in the input data. Un-
coupled deterministic problems can be derived to find each term in the expansion, the
previous term being needed to compute the next one. Each of these problems can be
solved approximately using for instance the finite element method (FEM). We derive a
priori and a posteriori error estimators in various norms for the error between the exact
solution and an approximation of a certain order. For instance for the first order approx-
imation, which requires the resolution of only one deterministic problem, we obtain an a posteriori
error estimator constituted of two computable parts, namely a part due to FE discretization (which
depends on the mesh size) and another part due to the uncertainties affecting the input data.
This estimator, easy and cheap to compute, can then be used for mesh adaptation to
balance the two sources of error.
We apply this method to several classes of problems. We first consider the linear elliptic problem
−div(a(x, ω)∇u(x, ω)) = f (x) (2)
where the random diffusion coefficient a depends in an affine way on a finite number of
independent random variables. The derivation of a priori and a posteriori error estimates in
various norms for the first order approximation u ≈ u0,h, with u0,h the continuous,
piecewise linear finite element approximation of u0, is detailed in [1]. The analysis
is
straightforwardly extended to higher order approximations and to some class of nonlinear problems.
We then consider the steady Navier-Stokes equations on randomly perturbed domains and, in
particular, the flow past a cylinder with perturbation of the center or the outer shape
of the cylinder. A stochastic mapping is introduced which transforms the original problem to PDEs
on a deterministic reference domain with random coefficients. Finally, we consider the heat
equation with random (Robin) boundary conditions. For this time-dependent problem, the a
posteriori error estimator contains a third term due
to time discretization.
CitacióGuignard, D.; Nobile, F.; Picasso, M. A posteriori error estimation for partial differential equations with small uncertainties. A: ADMOS 2015. CIMNE, 2015, p. 77-78.
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Admos2015-56-A POSTERIORI ERROR ESTIMATION.pdf | 105,7Kb | Visualitza/Obre |