Reduced Basis Error Bound Computation of Parameter-Dependent Navier–Stokes Equations by the Natural Norm Approach
- 1 January 2008
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Numerical Analysis
- Vol. 46 (4), 2039-2067
- https://doi.org/10.1137/060674181
Abstract
This work focuses on the a posteriori error estimation for the reduced basis method applied to partial differential equations with quadratic nonlinearity and affine parameter dependence. We rely on natural norms—local parameter-dependent norms—to provide a sharp and computable lower bound of the inf-sup constant. We prove a formulation of the Brezzi-Rappaz-Raviart existence and uniqueness theorem in the presence of two distinct norms. This allows us to relax the existence condition and to sharpen the field variable error bound. We also provide a robust algorithm to compute the Sobolev embedding constants involved in the error bound and in the inf-sup lower bound computation. We apply our method to a steady natural convection problem in a closed cavity, with a Grashof number varying from 10 to $10^7$.
Keywords
This publication has 21 references indexed in Scilit:
- A reduced basis element method for the steady Stokes problemESAIM: Mathematical Modelling and Numerical Analysis, 2006
- Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equationsComptes Rendus Mathematique, 2002
- An optimal control approach to a posteriori error estimation in finite element methodsActa Numerica, 2001
- Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient MethodSIAM Journal on Scientific Computing, 2001
- Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problemsComptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2000
- A Reduced-Order Method for Simulation and Control of Fluid FlowsJournal of Computational Physics, 1998
- PARAMETRIC FAMILIES OF REDUCED FINITE ELEMENT MODELS. THEORY AND APPLICATIONSMechanical Systems and Signal Processing, 1996
- On the Error Behavior of the Reduced Basis Technique for Nonlinear Finite Element ApproximationsZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1983
- Finite dimensional approximation of nonlinear problemsNumerische Mathematik, 1980
- Automatic choice of global shape functions in structural analysisAIAA Journal, 1978