Robust weak Galerkin finite element methods for linear elasticity with continuous displacement trace approximation
For researchers in computational mechanics, this provides a robust numerical method for nearly incompressible elasticity problems, though it is an incremental extension of existing WG methods.
This paper proposes new weak Galerkin finite element methods for linear elasticity that use continuous displacement trace approximation, achieving optimal and uniform error estimates independent of the Lamé constant λ, as confirmed by numerical experiments.
This paper proposes and analyzes a class of new weak Galerkin (WG) finite element methods for 2- and 3-dimensional linear elasticity problems. The methods use discontinuous piecewise-polynomial approximations of degrees $k(\geq 0)$ for the stress, $k+1$ for the displacement, and a continuous piecewise-polynomial approximation of degree $k+1$ for the displacement trace on the inter-element boundaries, respectively. After the local elimination of unknowns defined in the interior of elements, the WG methods result in SPD systems where the unknowns are only the degrees of freedom describing the continuous trace approximation. We show that the proposed methods are robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lamé constant $λ$. Numerical experiments confirm the theoretical results.