We discretize the Lagrange multiplier formulation of the obstacle problem by mixed and stabilized finite element methods. A priori and a posteriori error estimates are derived and numerically verified.

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Partial differential equations (PDEs)-such as the Navier-Stokes equations in fluid mechanics, the Maxwell equations in electromagnetism, and the Schrödinger equation in quantum mechanics-are the basic building blocks of modern physics and engineering. The finite element method (FEM) is a flexible computational technique for the discretization and solution of PDEs, especially in the case of complex spatial domains.Conceptually, the FEM transforms a time-independent (or temporally discretized) PDE into a system of linear equations Ax = b. scikit-fem is a lightweight Python library for the creation, or assembly, of the finite element matrix A and vector b. The user loads a computational mesh, picks suitable basis functions, and provides the PDE's weak formulation (Logg, Mardal, Wells, & others, 2012). This results in sparse matrices and vectors compatible with the SciPy (Virtanen et al., 2020) ecosystem.

We derive a novel and rigorous correction to the classical Reynolds lubrication approximation for fluids with viscosity depending upon the pressure. Our analysis shows that the pressure dependence of viscosity leads to additional nonlinear terms related to the shear-rate and arising from a non negligible cross-film pressure. We present a numerical comparison between the classical Reynolds equation and our modified equation and conclude that the modified equation leads to the prediction of higher pressures and viscosities in the flow domain.

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