A parameter-free and locking-free enriched Galerkin method of arbitrary order for linear elasticity

We propose a parameter-free and locking-free enriched Galerkin method of arbitrary order for solving the linear elasticity problem in both two and three space dimensions. Our method uses an approximation space that enriches the vector-valued continuous Galerkin space of order $k$ with some discontinuous piecewise polynomials. To the best of our knowledge, it extends the locking-free enriched Galerkin space in Yi et al. (2022) to high orders for the first time. Compared to the continuous Galerkin method, the proposed method is locking-free with only $k^{d-1}$ additional degree of freedom on each element. The parameter-free property of our method is realized by integrating the enriched Galerkin space into the framework of the modified weak Galerkin method. We rigorously establish the well-posedness of the method and provide optimal error estimates for the compressible case. Extensive numerical examples confirm both the accuracy and the locking-free property of the proposed method.