L. Machiels

Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems

Abstract We propose a new reduced-basis output bound method for the symmetric eigenvalue problem. The numerical procedure consists of two stages: the pre-processing stage, in which the reduced basis and associated functions are computed—“off-line”—at a prescribed set of points in parameter space; and the real-time stage, in which the approximate output of interest and corresponding rigorous error bounds are computed—“on-line”—for any new parameter value of interest. The real time calculation is very inexpensive as it requires only the solution or evaluation of very small systems. We introduce the procedure; prove the asymptotic bounding properties and optimal convergence rate of the error estimator; discuss computational considerations; and, finally, present corroborating numerical results.

A “flux-free” nodal Neumann subproblem approach to output bounds for partial differential equations

Abstract We present a “flux-free” nodal Neumann subproblem approach for the inexpensive computation of lower and upper bounds for output functionals of partial differential equations. The method resembles earlier nodal Dirichlet subproblem a posteriori approaches, except that the underlying partition of unity now appears explicitly in a modified residual; the latter ensures solvability of the corresponding Neumann problems, which, in turn, provides the requisite constant-free bounds. The new approach is considerably simpler to analyze and, more importantly, to implement, than previous hybrid-flux elemental Neumann subproblem techniques.

Output bounds for reduced-order approximations of elliptic partial differential equations

Abstract We present an a posteriori finite element procedure that provides inexpensive, rigorous, accurate, and constant-free lower and upper bounds for the error in the outputs – engineering quantities of interest – predicted by (Lagrangian) reduced-order approximations to coercive elliptic partial differential equations. The bound calculation requires (i) the reduced-order approximation for the primal and dual field variables, (ii) a lower bound for the minimum eigenvalue of the symmetric part of the operator, and (iii) the solution of purely local symmetric Neumann subproblems defined on small decoupled nodal overlapping patches. There are two critical components to the Neumann subproblems: a partition-of-unity attenuated local residual which eliminates hybrid fluxes from both the construction and analysis of the resulting estimators while simultaneously preserving the global bound property; and a L 2 -regularization term which provides stability despite the absence of local nodal equilibrium of the reduced-order primal and dual solutions. The estimator bounding property and optimal convergence rate (as the reduced-order basis is enriched) are proven, and corroborating numerical results are presented for two examples: a heat conduction fin (symmetric) problem; and a conjugate advection-diffusion/multi-material heat transfer (non-symmetric) problem.

A posteriori finite element bounds for output functionals of discontinuous Galerkin discretizations of parabolic problems

We present a Neumann-subproblem a posteriori finite element procedure for the efficient calculation of constant-free, sharp lower and upper estimators for linear functional outputs of parabolic equations discretized by a discontinuous Galerkin method in time. In space, a global coarse mesh and a decoupled fine mesh are used to compute the estimators which are shown to converge to the value of the output obtained for a global coupled fine mesh. We first formulate the bound procedure, with particular emphasis on the proof of the bounding properties. We then provide an illustrative numerical example: a problem of heat conduction in a composite material.

Finite Element Output Bounds for Parabolic Equations: Application to Heat Conduction Problems

We present a Neumann-subproblem a posteriori finite element procedure for the efficient calculation of rigorous, constant-free, sharp lower and upper estimators for linear functional outputs of parabolic equations discretized by a discontinuous Galerkin method in time. We first formulate the bound procedure; we then provide illustrative numerical examples for problems of unsteady heat conduction.