Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations
TL;DR: In this paper, the authors extended the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in volving.
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Abstract: In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.
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Citations
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References
A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations
TL;DR: It is shown that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced–basis approximation converges exponentially to the exact solution uniformly in parameter space, thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges.
Estimation of the error in the reduced basis method solution of nonlinear equations
TL;DR: The reduced basis error is shown to be dominated by an approximation error, which leads to error estimates for projection onto specific subspaced; for example, subspaces related to Taylor, Lagrange and discrete least-squares approximation.
A posteriori error estimation for reduced-basis approximation of parametrized elliptic coercive partial differential equations : “convex inverse” bound conditioners
TL;DR: A new class of improved bound conditioners is introduced: the critical innovation is the direct approximation of the parametric dependence of the inverse of the operator (rather than the operator itself); this helps accommodate higher-order effectivity constructions while simultaneously preserving on-line efficiency.