Numerical Analysis Seminar

Abstract: In this talk, we perform a posteriori error analysis for partial differential equations with uncertain input data characterized using random variables. Considering first small uncertainties, we use a perturbation approach expanding the solution of the problem with respect to a parameter ε that controls the amount of uncertainty. We derive residual-based a posteriori error estimates that control the two sources of error: the finite element discretization and the truncation in the expansion. The methodology is presented first on an elliptic equation with random coefficients and then on the steady-state Navier-Stokes equations on random domains. In the case of large uncertainties, we use instead the stochastic collocation method for the random space approximation. We present a residual-based a posteriori error estimate that provides an upper bound for the total error, which is composed of the finite element and the stochastic collocation errors. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm. Several numerical examples are presented to illustrate the theoretical findings.

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