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

TL;DR

The paper develops a unified Least-Squares Finite Element framework for nonlinear PDEs by recasting the system as a residual minimization problem with the objective $F(u)=|R(u)+f|^2$. It derives a Gauss-Newton iteration to minimize a linearized LS functional, proves coercivity and continuity of the LS operator, and shows that $F$ provides a reliable and efficient a posteriori error estimator to drive adaptive mesh refinement. The methodology is demonstrated on nonlinear Poisson problems with variable or discontinuous coefficients, ReLU activations, nonlinear Saint-Venant–Kirchhoff elasticity, and sea-ice momentum balance, highlighting norm-equivalence, convergence guarantees, and effective adaptivity. Across these applications, the framework supports sharp error control and optimal convergence, underscoring its theoretical robustness and practical potential for complex nonlinear problems.

Abstract

This paper presents a unified Least-Squares framework for solving nonlinear partial differential equations by recasting the governing system as a residual minimisation problem. A Least-Squares functional is formulated and the corresponding Gauss-Newton method derived, which approximates simultaneously primal and dual 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, nonlinear elasticity based on the Saint-Venant Kirchhoff model and sea-ice dynamics.

Figures (2)

• Figure 1: Solution of the stationary heat equation on an L-shaped domain using $u_\mathcal{T} \in \left(\mathrm{P}_1\right)^2$ and $\sigma_\mathcal{T} \in \left(\mathrm{RT}_{0}\right)^2$: The adapted mesh after six refinement cycles (a) and the corresponding solution (b).
• Figure 2: Cooks membrane: The boundary value problem with initial mesh (a) and the mesh after eight adaptive refinement cycles (b).

Theorems & Definitions (21)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Lemma 4.1
• proof