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This paper develops a new recovery-based linear C0 finite element method for approximating the weak solution of a fourth-order singularly perturbed Monge-Ampère equation, which is known as the vanishing moment approximation of the Monge-Ampère equation. The proposed method uses a gradient recovery technique to define a discrete Laplacian for a given linear C0 finite element function (offline), the discrete Laplacian is then employed to discretize the biharmonic operator appeared in the equation. It is proved that the proposed C0 linear finite element method has a unique solution using a fixed point argument and the corresponding error estimates are derived in various norms. Numerical experiments are also provided to verify the theoretical error estimates and to demonstrate the efficiency of the proposed recovery-based linear C0 finite element method. [ABSTRACT FROM AUTHOR]

In this paper, we propose two systematic strategies to recover the gradient on the boundary of a domain. The recovered gradient has comparable superconvergent property on the boundary as that in the interior of the domain. This superconvergence property has been validated by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we discuss locking and robustness of the finite element method for a model circular arch problem. It is shown that in the primal variable (i.e., the standard displacement formulation), the p-version is free from locking and uniformly robust with order $p^{-k}$ and hence exhibits optimal rate of convergence. On the other hand, the h-version shows locking of order $h^{-2}$ , and is uniformly robust with order $h^{p-2}$ for $p>2$ which explains the fact that the quadratic element for some circular arch problems suffers from locking for thin arches in computational experience. If mixed method is used, both the h-version and the p-version are free from locking. Furthermore, the mixed method even converges uniformly with an optimal rate for the stress. [ABSTRACT FROM AUTHOR]