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A least-squares finite element reduced basis method. (English) Zbl 1481.65221

The paper introduces a reduced basis method for parametrized linear elliptic partial differential equations (PDEs) in a least-squares finite element framework. A rigorous and reliable error estimate is developed, and it is shown to bound the error with respect to the exact solution of the PDE, in contrast to estimates that measure error with respect to a finite-dimensional approximation. It is shown that the first-order formulation of the least-squares finite element is a key ingredient. The method is demonstrated using numerical examples. Although the bound on the effectivity is not sharp, it is still accurate.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
35J15 Second-order elliptic equations

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Firedrake; redbKIT
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References:

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