Superconvergence of gradient recovery on deviated discretized manifolds

10/30/2019

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by   Guozhi Dong, et al.

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This paper addresses open questions proposed by Wei, Chen and Huang [ SIAM J. Numer. Anal., 48(2010), pp. 1920–1943]. Mainly two questions arise there: (i) How to design gradient recovery algorithms given no exact information of the underline surfaces; (ii) Whether the superconvergence still holds when the vertices of the triangle mesh have O(h^2) deviation to the underline exact surfaces. We positively answer both questions. For the first, we propose a family of isoparametric gradient recovery schemes, which turn out to be nature generalizations of classical recovery methods from the Euclidean domain to manifolds. For the second, we prove a property called geometric supercloseness condition, which subsequently leads to the desired superconvergence result. Numerical results are documented for verification.