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Isogeometric analysis and proper orthogonal decomposition for parabolic problems. (English) Zbl 1380.65295

Summary: We investigate the combination of Isogeometric Analysis (IGA) and proper orthogonal decomposition (POD) based on the Galerkin method for model order reduction of linear parabolic partial differential equations. For the proposed fully discrete scheme, the associated numerical error features three components due to spatial discretization by IGA, time discretization with the \(\theta \)-scheme, and eigenvalue truncation by POD. First, we prove a priori error estimates of the spatial IGA semi-discrete scheme. Then, we show stability and prove a priori error estimates of the space-time discrete scheme and the fully discrete IGA-\(\theta \)-POD Galerkin scheme. Numerical tests are provided to show efficiency and accuracy of NURBS-based IGA for model order reduction in comparison with standard finite element-based POD techniques.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65D07 Numerical computation using splines
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs

Software:

GeoPDEs
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Full Text: DOI Link

References:

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