Analysis of least squares finite element methods for a parameter-dependent first-order system

A parameter-dependent first-order system arising from elasticity problems is introduced. The system corresponds to the 2D isotropic elasticity equations under a stress-pressure-displacement formulation in which the nonnegative parameter measures the material compressibility for the elastic body. Standard and weighted least squares finite element methods are applied to this system, and analyses for different values of the parameter are performed in a unified manner. The methods offer certain advantages such as they need not satisfy the Babuska-Brezzi condition, a single continuous piecewise polynomial space can be used for the approximation of all the unknowns, the resulting algebraic system is symmetric and positive definite, accurate approximations of all the unknowns can be obtained simultaneously, and, especially, computational results do not exhibit any significant numerical locking as the parameter tends to zero which corresponds to the incompressible elasticity problem (or equivalently, the Stokes problem). With suitable boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H¹-norm and in the L²-norm for all the unknowns. Numerical experiments with various values of the parameter are given to demonstrate the theoretical estimates.