Iterative energy reduction Galerkin methods and variational adaptivity
Pascal Heid, Thomas P. Wihler
TL;DR
This work addresses the numerical approximation of critical points of an energy functional $E$ in variational PDEs by introducing an energy-guided iterative linearized Galerkin (ILG) framework. A two-tier approach is developed: (i) an inner ILG solver that guarantees energy reduction at every step, with a computable discrete residual guiding stopping, and (ii) a global variational adaptivity strategy that enriches Galerkin spaces based on local energy reductions to achieve convergence as the discretization is refined. The authors provide a rigorous convergence theory under structural assumptions (A1)–(A3) and (T) and illustrate variational adaptivity in a finite element setting, verified on nonlinear diffusion–reaction models including semilinear and quasilinear cases. Numerical experiments demonstrate robust energy decay, competitive convergence rates, and effective mesh refinement driven solely by energy considerations, highlighting the method’s practicality when standard a posteriori estimators are unavailable. Overall, the paper offers a cohesive energy-centered framework that unifies iterative solvers and energy-aware adaptivity for a broad class of nonlinear variational problems with strong potential for practical impact in physics and engineering computations.
Abstract
Critical points of energy functionals, which are of broad interest, for instance, in physics and chemistry, in solid and quantum mechanics, in material science, or in general diffusion-reaction models arise as solutions to the associated Euler-Lagrange equations. While classical computational solution methods for such models typically focus solely on the underlying partial differential equations, we propose an approach that also incorporates the energy structure itself. Specifically, we examine (linearized) iterative Galerkin discretization schemes that ensure energy reduction at each step, and utilize the computable discrete residual to determine an appropriate stopping point. Additionally, we provide necessary conditions, which are applicable to a wide class of problems, that guarantee convergence to critical points of the PDE as the discrete spaces are enriched. Moreover, in the specific context of finite element discretizations, we present a very generally applicable adaptive mesh refinement strategy - the so-called variational adaptivity approach - which, rather than using classical a posteriori estimates, is based on exploiting local energy reductions. The theoretical results are validated for several computational experiments in the context of nonlinear diffusion-reaction models, thereby demonstrating the effectiveness of the proposed scheme.