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

TL;DR

This paper introduces a novel fully iterative adaptive energy-based algorithm for solving strongly convex elliptic problems, using energy reduction principles to guide refinement and stopping criteria, demonstrated to achieve optimal convergence.

Contribution

The approach uniquely employs energy reduction indicators for adaptive refinement and solver stopping, differing from traditional a posteriori estimator-based methods.

Findings

Achieves optimal convergence rates in numerical experiments.

Effectively refines finite element spaces based on local energy reductions.

Demonstrates applicability to second-order semilinear elliptic models.

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.