This paper analyzes lowest-order discontinuous Petrov--Galerkin (dPG) finite element methods (FEM) for the Navier--Lame equations with different norms and side restrictions. The focus is on the direct proof of a discrete inf-sup condition for a low-order test-search space. The low-order finite element spaces in the ultraweak formulation involve the piecewise constant and affine ansatz functions and discontinuous piecewise affine test functions in two and three space dimensions with Neumann boundary conditions or the pure Dirichlet problem. Those lowest-order discretizations for linear elasticity allow for a direct proof of the discrete inf-sup condition and a complete a priori and a posteriori error analysis which is robust in the incompressible limit as $\lambda \rightarrow \infty$. Numerical experiments with uniform and adaptive mesh-refinements investigate $\lambda$-robustness and confirm that one scheme is locking-free.
['Carsten Carstensen', 'Friederike Hellwig']
Low-Order Discontinuous Petrov--Galerkin Finite Element Methods for Linear Elasticity