Convergence analysis of the Newton solver and its interplay with adaptivity

Develop a thorough convergence analysis for the Newton method employed to solve the Euler–Lagrange equations of the proposed discontinuous Petrov–Galerkin (DPG) plus least-squares minimum-residual discretization of the semilinear elliptic problem, and ascertain the interaction between the Newton stopping criterion and the adaptive mesh refinement strategy to ensure reliable and efficient convergence behavior.

Background

The paper proposes a minimum-residual finite-element scheme that combines a DPG formulation for the linearized part of a semilinear elliptic problem with least-squares constraints to capture the nonlinear relations. The resulting nonlinear discrete system is solved with a Newton method.

While the method is validated numerically, a rigorous convergence theory for the Newton iterations in this specific DPG–least-squares setting, and its coupling with adaptive mesh refinement (including how to set and scale the stopping criterion during refinement), is not provided here and is explicitly deferred. Establishing such a theory is important for guaranteeing robustness and efficiency of the algorithm in practice.