Iterative energy reduction Galerkin methods and variational adaptivity

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.