Colloquium October 16
Date: Friday, October 16
Time: 16:00 to 17:00
Place: 455 McBryde (Commons Room)
Speaker: Timothy Barth of the NASA Ames Research Center
Title: Error Representation for Steady-State and Time-Dependent Compressible Flow Calculations
Abstract: For better or worse, our physical world is constantly evolving in time. This leads to a multitude of challenges associated with identifying, quantifying, and controlling numerical errors in complex time dependent numerical simulations. For example, in turbulent flow simulations it is well known that the control of pointwise solution errors quickly becomes intractable as the flow Reynolds number increases but the control of errors occurring in statistics and space-time integrated quantities may still be tractable [1]. In this seminar, we consider the representation and control of numerical solution errors in space-time FE methods using standard duality techniques as succinctly described in [2,3]. For computed output quantities of interest that are mathematically described as functionals, this technique exploits the precise relationship between functional errors and weighted combinations of computable element residuals. The associated non-perturbative theory elucidates the exact form of these residual weights given the finite-dimensional primal solution and the infinite-dimensional dual solution.
Numerous works [4,5,6] have applied this technology to achieve error control for steady-state finite element computations. We demonstrate this procedure for steady-state finite element computations of compressible flow past aerodynamic bodies. A more demanding task is the representation and control of numerical errors for time-dependent finite element calculations. This task introduces a multitude of computational difficulties surrounding the approximation of the locally linearized dual (backwards in time) problem. For example, the resources required to store the primal (forward in time) solution may become prohibitively large for long time integrations so that alternative strategies that trade storage for computation must be employed. We then examine the representation of functional error via dual problems for cylinder flow at low Reynolds number when the flow becomes periodic in time with the goal of studying the deterioration in these dual problems with increasing Reynolds number. We then discuss the prospects for error control of general time dependent Navier-Stokes flow computations.
[1] J. Hoffman and C. Johnson,"Adaptive Finite Element Methods in
Incompressible Fluid Flow", LNCSE, Vol. 25, Springer-Verlag Pub,
2002.
[2] R. Becker and R. Rannacher,"An Optimal Control Approach to
A-Posteriori Error Estimation in Finite Element Methods, Acta
Numerica, 2001.
[3] K. Ericksson, D. Estep, P. Hansbo and C. Johnson, Computational
Differential Equations, Cambridge Press., 1996.
[4] T. Barth and M. Larson, "A-Posteriori Error Estimation for
Adaptive Discontinuous Galerkin Approximations of Hyperbolic
Systems", LNCSE, Vol. 11, 1999.
[5} J. T. Oden and S. Prudhomme, "Goal-Oriented Error Estimation and
Adaptivity for the Finite Element Method", TICAM 99-015, 1999.
[6] R. Hartman and P. Houston, "Symmetric Interior Penalty DG Methods for the Compressible Navier-Stokes Equations II: Goal-Oriented A Posteriori Error Estimation", Int. J. Numer. Anal. Model., 2006.
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