Analysis of Least Squares Finite Element Methods for A Parameter-Dependent First-Order System

A parameter-dependent rst-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 nite element methods are applied to this system, and analyses for diierent values of the parameter are performed in a uniied manner. The methods ooer certain advantages such as they need not satisfy the Babu ska-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 deenite, accurate approximations of all the unknowns can be obtained simultaneously , and, especially, computational results do not exhibit any signiicant numerical locking as the parameter tends to zero which corresponds to the incom-pressible elasticity problem (or equivalently, the Stokes problem). With suitable boundary conditions, it is shown that both methods achieve optimal rates of convergence in the H 1-norm and in the L 2-norm for all the unknowns. Numerical experiments with various values of the parameter are given to demonstrate the theoretical estimates.