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.

翻译：能量泛函的临界点在物理学、化学、固体与量子力学、材料科学以及一般扩散-反应模型等诸多领域具有广泛的研究意义，其解对应于相应的欧拉-拉格朗日方程。传统的此类模型计算方法通常仅关注底层的偏微分方程，本文提出一种同时融合能量结构本身的新途径。具体而言，我们研究能确保每一步迭代均实现能量缩减的（线性化）迭代Galerkin离散化格式，并利用可计算的离散残差确定合适的迭代终止点。此外，我们建立了适用于广泛问题类别的充分条件，保证在离散空间不断加密时算法收敛至偏微分方程的临界点。特别地，在有限元离散化框架下，我们提出一种具有普适性的自适应网格细化策略——即所谓变分自适应方法——该方法摒弃经典的后验估计，转而基于局部能量缩减原理构建。通过非线性扩散-反应模型的多组数值实验验证了理论结果，从而证明了所提方案的有效性。