Least-Squares Finite Element Methods for nonlinear problems: A unified framework

Abstract: This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimization problem. A Least-Squares functional is formulated and the corresponding Gauss-Newton iterative method derived, which approximates simultaneously primal and stress-like variables. We derive conditions under which the Least-Squares functional is coercive and continuous in an appropriate solution space, and establish convergence results while demonstrating that the functional serves as a reliable a posteriori error estimator. This inherent error estimation property is then exploited to drive adaptive mesh refinement across a variety of problems, including the stationary heat equation with either temperature-dependent or discontinuous conductivity or a ReLU-type nonlinearity, nonlinear elasticity based on the Saint Venant-Kirchhoff model and sea-ice dynamics.