An adaptive Stochastic Finite Elements approach based on Newton–Cotes quadrature in simplex elements

Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws

“…Although these deterministic methods show some promise, they suffer from the disadvantage that they are highly intrusive: existing codes for computing deterministic solutions of balance (conservation) laws need to be completely reconfigured for implementation of the gPC based stochastic Galerkin methods. An alternative class of methods for quantifying uncertainty in PDEs are the stochastic collocation methods, see [47] for a general review and [28,46] for modifications of these methods near discontinuities. Stochastic collocation methods are nonintrusive and easier to parallelize than the gPC based stochastic Galerkin methods.…”

Higher-Order Newton-Cotes Formulas

“…The closed forms are those where the data points at the beginning and end of the limits of integration are known. The open forms have integration limits that extend beyond the range of the data (Witteveen et al, 2009).…”

Unsteady Adaptive Stochastic Finite Elements for Quantification of Uncertainty in Time-Dependent Simulations

“…The paper is concluded in section 5. The consistence of the approaches has been verified by comparison of results for analytical test problems with those of Monte Carlo simulations in previous studies [18,19,20,21].…”

“…For a robust interpolation an alternative Adaptive Stochastic Finite Elements formulation is proposed based on Newton-Cotes quadrature in simplex elements [18], which preserves monotonicity and extrema of the low number of samples. The resulting Unsteady Adaptive Stochastic Finite Elements (UASFE) approach [21] can be applied to problems in which the phase of the oscillatory samples is well-defined.…”

“…For oscillatory time-dependent responses a Fourier Chaos basis [13] has recently been proposed. Multi-element Adaptive Stochastic Finite Elements (ASFE) methods [14,15,16,17,18] have been developed in order to approximate discontinuities by a more robust piecewise polynomial interpolation of the samples.…”