New Nonconforming Elements for Linear Strain Gradient Elastic Model

Based on a new H$^2-$Korn's inequality, we propose new nonconforming elements for the linear strain gradient elastic model. The first group of elements are H$^1-$conforming but H$^2-$nonconforming. The tensor product NTW element [Tai:2001] and the tensor product Specht triangle are two typical representatives. The second element is based on Morley's triangle with a modified elastic strain energy. We proved new interpolation error estimates for all these elements, which are key to prove uniform rates of convergence for the proposed elements. Numerical results are reported and they are consistent with the theoretical prediction.