114 research outputs found
Adjoint recovery of superconvergent functionals from PDE approximations
Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations
Exploiting Superconvergence Through Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering
There has been much work in the area of superconvergent error analysis for finite element and discontinuous Galerkin (DG) methods. The property of superconvergence leads to the question of how to exploit this information in a useful manner, mainly through superconvergence extraction. There are many methods used for superconvergence extraction such as projection, interpolation, patch recovery and B-spline convolution filters. This last method falls under the class of Smoothness-Increasing Accuracy-Conserving (SIAC) filters. It has the advantage of improving both smoothness and accuracy of the approximation. Specifically, for linear hyperbolic equations it can improve the order of accuracy of a DG approximation from k + 1 to 2k + 1, where k is the highest degree polynomial used in the approximation, and can increase the smoothness to k − 1. In this article, we discuss the importance of overcoming the mathematical barriers in making superconvergence extraction techniques useful for applications, specifically focusing on SIAC filtering
Self-Adaptive Methods for PDE
[no abstract available
Linearization Errors in Discrete Goal-Oriented Error Estimation
Goal-oriented error estimation provides the ability to approximate the
discretization error in a chosen functional quantity of interest. Adaptive mesh
methods provide the ability to control this discretization error to obtain
accurate quantity of interest approximations while still remaining
computationally feasible. Traditional discrete goal-oriented error estimates
incur linearization errors in their derivation. In this paper, we investigate
the role of linearization errors in adaptive goal-oriented error simulations.
In particular, we develop a novel two-level goal-oriented error estimate that
is free of linearization errors. Additionally, we highlight how linearization
errors can facilitate the verification of the adjoint solution used in
goal-oriented error estimation. We then verify the newly proposed error
estimate by applying it to a model nonlinear problem for several quantities of
interest and further highlight its asymptotic effectiveness as mesh sizes are
reduced. In an adaptive mesh context, we then compare the newly proposed
estimate to a more traditional two-level goal-oriented error estimate. We
highlight that accounting for linearization errors in the error estimate can
improve its effectiveness in certain situations and demonstrate that localizing
linearization errors can lead to more optimal adapted meshes
Sharp error estimates for a discretisation of the 1D convection/diffusion equation with Dirac initial data
This paper derives sharp l and l1 estimates of the error arising from an explicit approximation of the constant coefficient 1D convection/diffusion equation with Dirac initial data. The analysis embeds the discrete equations within a semi-discrete system of equations which can be solved by Fourier analysis. The error estimates are then obtained through asymptotic approximation of the integrals resulting from the inverse Fourier transform. this research is motivated by the desire to prove convergence of approximations to adjoint partial differential equations
A weighted reduced basis method for parabolic PDEs with random data
This work considers a weighted POD-greedy method to estimate statistical
outputs parabolic PDE problems with parametrized random data. The key idea of
weighted reduced basis methods is to weight the parameter-dependent error
estimate according to a probability measure in the set-up of the reduced space.
The error of stochastic finite element solutions is usually measured in a root
mean square sense regarding their dependence on the stochastic input
parameters. An orthogonal projection of a snapshot set onto a corresponding POD
basis defines an optimum reduced approximation in terms of a Monte Carlo
discretization of the root mean square error. The errors of a weighted
POD-greedy Galerkin solution are compared against an orthogonal projection of
the underlying snapshots onto a POD basis for a numerical example involving
thermal conduction. In particular, it is assessed whether a weighted POD-greedy
solutions is able to come significantly closer to the optimum than a
non-weighted equivalent. Additionally, the performance of a weighted POD-greedy
Galerkin solution is considered with respect to the mean absolute error of an
adjoint-corrected functional of the reduced solution.Comment: 15 pages, 4 figure
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