Least-squares variational principles and the finite element method: theory, formulations, and models for solid and fluid mechanics

We consider the application of least-squares variational principles and the finite element method to the numerical solution of boundary value problems arising in the fields of solid and fluid mechanics. For many of these problems least-squares principles offer many theoretical and computational advantages in the implementation of the corresponding finite element model that are not present in the traditional weak form Galerkin finite element model. For instance, the use of least-squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise. Furthermore, the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix, allowing the use of robust and fast iterative methods for its solution. We find that the use of high p-levels is beneficial in least-squares based finite element models and present guidelines to follow when a low p-level numerical solution is sought. Numerical examples in the context of incompressible and compressible viscous fluid flows, plate bending, and shear-deformable shells are presented to demonstrate the merits of the formulations.