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**An optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems.**
*(English)*
Zbl 1161.76032

Summary: We combine the proper orthogonal decomposition (POD) with Petrov-Galerkin least squares mixed finite element (PLSMFE) method to derive an optimizing reduced PLSMFE formulation for non-stationary conduction-convection problems. Error estimates between the optimizing reduced PLSMFE solutions based on POD and classical PLSMFE solutions are presented. The optimizing reduced PLSMFE formulation can circumvent the constraint of Babuška-Brezzi condition, so that the combination of finite element subspaces can be chosen freely and allow optimal-order error estimates to be obtained. Numerical examples show that the errors between the optimizing reduced PLSMFE solutions and the classical PLSMFE solutions are consistent with theoretical results. Moreover, they also show the feasibility and efficiency of the POD method.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76R99 | Diffusion and convection |

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\textit{Z. Luo} et al., Int. J. Numer. Methods Fluids 60, No. 4, 409--436 (2009; Zbl 1161.76032)

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