A fully iterative adaptive energy-based approach for monotone elliptic problems

Abstract

We present a fully iterative adaptive algorithm for the numerical minimization of strongly convex energy functionals in Hilbert spaces. The proposed approach, which we first present in abstract form, generates a hierarchical sequence of adaptively refined finite-dimensional approximation spaces and employs a (nonlinear) conjugate gradient (CG) method to compute suitable approximations on each space. A core novelty of our approach is that all components of the algorithm are consistently driven by energy reduction principles rather than by classical a posteriori estimators. In particular, adaptive refinement is steered by local energy reduction indicators which aim to construct subsequent approximation spaces in a way that attains the largest potential decrease in energy. Likewise, the stopping criteria for the iterative solver are based on either relative or averaged energy reductions on each subspace. As a concrete realization, we present a concise implementation for $\mathbb{P}_1$ finite element discretizations of second-order semilinear elliptic diffusion-reaction models, where the local indicators driving the element refinements are computed based on edge-wise energy reductions. Numerical experiments demonstrate that the resulting scheme achieves optimal convergence for various benchmark problems in two-dimensional polygonal domains.