On goal-oriented error estimation and adaptivity for nonlinear systems with uncertain data and application to flow problems

The objective of this work is to develop a posteriori error estimates and adaptive strategies for the numerical solution to nonlinear systems of partial differential equations with uncertain data. Areas of application cover problems in fluid mechanics including a Bayesian model selection study of turbulence comparing different uncertainty models. Accounting for uncertainties in model parameters may significantly increase the computational time when simulating complex problems. The premise is that using error estimates and adaptively refining the solution process can reduce the cost of such simulations while preserving their accuracy within some tolerance. New insights for goal-oriented error estimation for deterministic nonlinear problems are first presented. Linearization of the adjoint problems and quantities of interest introduces higher-order terms in the error representation that are generally neglected. Their effects on goal-oriented adaptive strategies are investigated in detail here. Contributions on that subject include extensions of well-known theoretical results for linear problems to the nonlinear setting, computational studies in support of these results, and an extensive comparative study of goal-oriented adaptive schemes that do, and do not, include the higher-order terms. Approaches for goal-oriented error estimation for PDEs with uncertain coefficients have already been presented, but lack the capability of distinguishing between the different sources of error. A novel approach is proposed here, that decomposes the error estimate into contributions from the physical discretization and the uncertainty approximation. Theoretical bounds are proven and numerical examples are presented to verify that the approach identifies the predominant source of the error in a surrogate model. Adaptive strategies, that use this error decomposition and refine the approximation space accordingly, are designed and tested. All methodologies are demonstrated on benchmark flow problems: Stokes lid-driven cavity, 1D Burger’s equation, 2D incompressible flows at low Reynolds numbers. The procedure is also applied to an uncertainty quantification study of RANS turbulence models in channel flows. Adaptive surrogate models are constructed to make parameter uncertainty propagation more efficient. Using surrogate models and adaptivity in a Bayesian model selection procedure, it is shown that significant computational savings can be gained over the full RANS model while maintaining similar accuracy in the predictions.