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We present a novel energy-based numerical analysis of semilineardiffusion-reaction boundary value problems. Based on a suitable variationalsetting, the proposed computational scheme can be seen as an energyminimisation approach. More specifically, this procedure aims to generate asequence of numerical approximations, which results from the iterative solutionof related (stabilised) linearised discrete problems, and tends to a localminimum of the underlying energy functional. Simultaneously, thefinite-dimensional approximation spaces are adaptively refined; this isimplemented in terms of a new mesh refinement strategy in the context of finiteelement discretisations, which again relies on the energy structure of theproblem under consideration, and does not involve any a posteriori errorindicators. In combination, the resulting adaptive algorithm consists of aniterative linearisation procedure on a sequence of hierarchically refineddiscrete spaces, which we prove to converge towards a solution of thecontinuous problem in an appropriate sense. Numerical experiments demonstratethe robustness and reliability of our approach for a series of examples.

A numerical energy minimisation approach for semilinear diffusion-reaction boundary value problems based on steady state iterations