Interior Penalty Mixed Finite Element Methods of Any Order in Any Dimension for Linear Elasticity with Strongly Symmetric Stress Tensor
This work provides a unified framework for constructing stable mixed finite elements of any order in any dimension for linear elasticity, addressing a known bottleneck in the field.
The authors propose two classes of mixed finite elements for linear elasticity that achieve optimal error estimates for both displacement and stress using interior penalty methods, with numerical validation in 2D and 3D.
We propose two classes of mixed finite elements for linear elasticity of any order, with interior penalty for nonconforming symmetric stress approximation. One key point of our method is to introduce some appropriate nonconforming face-bubble spaces based on the local decomposition of discrete symmetric tensors, with which the stability can be easily established. We prove the optimal error estimate for both displacement and stress by adding an interior penalty term. The elements are easy to be implemented thanks to the explicit formulations of its basis functions. Moreover, the methods can be applied to arbitrary simplicial grids for any spatial dimension in a unified fashion. Numerical tests for both 2D and 3D are provided to validate our theoretical results.