The Gradient Discretisation Method For Variational Inequalities

The purpose of this thesis is to develop a gradient discretisation method for elliptic and parabolic, linear and non-linear, variational inequalities. The gradient discretisation method is a framework which enables a unified convergence analysis of many different methods – such as finite elements (conforming, non-conforming and mixed) and finite volumes methods – for 2nd order diffusion equations. Using the gradient discretisation method framework, we perform the numerical analysis of variational inequalities. We first establish error estimates for numerical approximations of linear elliptic variational inequalities. Using compactness techniques, we prove the convergence of numerical schemes for non-linear elliptic variational inequalities based on Leray–Lions operators. We also show the uniform-in-time convergence for linear parabolic variational inequalities. As numerical applications of this framework, we design, analyse and test the hybrid mimetic mixed (HMM) method for variational inequalities.