| As methods for the deterministic solutions of differential equations continue to mature, there is intensive interest in uncertainty quantification for more realistic mathematical models, where the physical applications are affected by uncertainty in the input data, such as model parameters, boundary/initial conditions, forcing terms, etc. In this work, a non-sampling method called "multi-element generalized polynomial chaos (ME-gPC)" is developed to study uncertainty quantification. In particular, adaptivity is coupled with this method to control simulation errors.; The Karhuen-Loeve (K-L) expansion, an efficient technique for modeling of random inputs, is first presented, where we focus on fast numerical algorithms for the associated eigenvalue problem, i.e., the key part of the K-L expansion. A sharp error estimate for the fast Gauss transform is developed and used to propose a fast eigenvalue solver for Gaussian-type covariance kernels, which yields a saving up to three orders of magnitude in time and memory.; The algorithm of ME-gPC is next presented. ME-gPC is based on the decomposition of random space and spectral expansions. In each random element, a new random variable subject to a conditional probability density function (PDF) is defined. Orthogonal polynomials weighted by the conditional PDF are numerically constructed fast and accurately. The hp-convergence ME-gPC is proved if the function has enough regularity in the parametric space.; Adaptive ME-gPC algorithms for ODEs and PDEs with random coefficients are subsequently developed. For ODEs, a heuristic adaptivity criterion is developed based on the decay rate of coefficients of local polynomial chaos expansions. For PDEs, a rigorous adaptivity criterion is developed based on a posteriori error estimators. In particular, a reduced space is constructed to reduce the cost of solving the error equations.; Two fundamental problems of polynomial chaos methods are next studied: long-term integration and non-Gaussian random inputs. The performance of gPC and ME-gPC is compared for a hyperbolic equation with a random transport velocity. It is shown that ME-gPC can extend the valid integration time for a given accuracy by employing h-convergence. A methodology is developed for polynomial chaos methods to deal with non-Gaussian random inputs more efficiently.; The developed algorithms are then applied to physical applications in fluid mechanics including heat conduction in a 3D electric chip, heat convection in a grooved channel and noisy flow past a stationary circular cylinder.; In addition to the ME-gPC method, a study on the flux-type a posteriori error estimation for spectral/hp finite element methods is included in this work. Explicit fluxes for the two-dimensional case are given. A more general approach is proposed to obtain equilibrated fluxes for the local pure Neumann problems by minimizing a weighted target function. |
