The displacement and gradient of a finite element solution are of primary interest and a high accuracy displacement and gradient approximation are always desirable in scientific computing. In this paper, we introduce a new state of the art way to reconstruct a high accuracy approximated gradient. The proposed method inherits the advantages of both the extrapolation technique and the interpolation post-processing technique. We theoretically justify the fifth-order local ultraconvergence of the post-processing gradient. In the meanwhile, an ultraconvergent displacement can be obtained from this post-processing procedure. The theoretical result is numerically verified and validated.

中文翻译：

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二次元局部超收敛

有限元解的位移和梯度是主要关注点，并且在科学计算中始终需要高精度的位移和梯度近似。在本文中，我们介绍了一种新的最先进的方法来重建高精度近似梯度。所提出的方法继承了外推技术和内插后处理技术的优点。我们从理论上证明了后处理梯度的五阶局部超收敛。同时，可以从这个后处理过程中获得超收敛位移。理论结果得到了数值验证和验证。