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A weakly over-penalized symmetric interior penalty method is applied to solve elliptic eigenvalue problems. We derive a posteriori error estimator of residual type, which proves to be both reliable and efficient in the energy norm. Some numerical tests are provided to confirm our theoretical analysis.
Cu/Sn alloy matrix composites reinforced with β-Si3N4 whiskers (β-Si3N4w/Cu/Sn) were prepared by powder metallurgy method with the aim of improving the mechanical property. Although the β-Si3N4 whisker additions played a side effect on the densification of the composites, the mechanical property was significantly improved. The highest bending strength of β-Si3N4w/Cu/Sn reached up to 353 MPa, and the highest hardness of composites was also improved to 97.13 HRF. β-Si3N4 whiskers could induce the improved work of fracture and grain refinement of composites and thus brought about the enhancement of mechanical performance. However, excessively high β-Si3N4 whisker content was not beneficial to the high mechanical property because of the low densification and the severe aggregation of whiskers. The β-Si3N4w/Cu/Sn still had a quite high thermal conductivity. As a result, the β-Si3N4w/Cu/Sn could be a promising material applied in a severe environment.
In this paper, we propose and analyze the interior penalty discontinuous Galerkin method for H(div)-elliptic problem. An optimal a priori error estimate in the energy norm is proved. In addition, a residual-based a posteriori error estimator is obtained. The estimator is proved to be both reliable and efficient in the energy norm. Some numerical testes are presented to demonstrate the effectiveness of our method.
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