scholarly journals A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations

2003 ◽  
Author(s):  
Karen Veroy ◽  
Christophe Prud'homme ◽  
Dimitrios Rovas ◽  
Anthony Patera
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Bounds ◽  
Posteriori Error ◽  
Elliptic Partial Differential Equations ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
Nonlinear Elliptic ◽  
A Posteriori Error Bounds

CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (03) ◽  
pp. 1150015 ◽  
Author(s):  
MARTIN A. GREPL
Keyword(s):  
Error Bounds ◽  
Linear Time ◽  
Posteriori Error ◽  
Parabolic Partial Differential Equations ◽  
Coefficient Function ◽  
Time Varying ◽  
A Posteriori ◽  
Reduced Basis ◽  
A Posteriori Error ◽  
A Posteriori Error Bounds

We present reduced basis approximations and associated a posteriori error bounds for parabolic partial differential equations involving (i) a nonaffine dependence on the parameter and (ii ) a nonlinear dependence on the field variable. The method employs the Empirical Interpolation Method in order to construct "affine" coefficient-function approximations of the "nonaffine" (or nonlinear) parametrized functions. We consider linear time-invariant as well as linear time-varying nonaffine functions and introduce a new sampling approach to generate the function approximation space for the latter case. Our a posteriori error bounds take both error contributions explicitly into account — the error introduced by the reduced basis approximation and the error induced by the coefficient function interpolation. We show that these bounds are rigorous upper bounds for the approximation error under certain conditions on the function interpolation, thus addressing the demand for certainty of the approximation. As regards efficiency, we develop an offline–online computational procedure for the calculation of the reduced basis approximation and associated error bound. The method is thus ideally suited for the many-query or real-time contexts. Numerical results are presented to confirm and test our approach.

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