  We present a novel energy-based numerical analysis of semilinear
diffusion-reaction boundary value problems. Based on a suitable variational
setting, the proposed computational scheme can be seen as an energy
minimisation approach. More specifically, this procedure aims to generate a
sequence of numerical approximations, which results from the iterative solution
of related (stabilised) linearised discrete problems, and tends to a local
minimum of the underlying energy functional. Simultaneously, the
finite-dimensional approximation spaces are adaptively refined; this is
implemented in terms of a new mesh refinement strategy in the context of finite
element discretisations, which again relies on the energy structure of the
problem under consideration, and does not involve any a posteriori error
indicators. In combination, the resulting adaptive algorithm consists of an
iterative linearisation procedure on a sequence of hierarchically refined
discrete spaces, which we prove to converge towards a solution of the
continuous problem in an appropriate sense. Numerical experiments demonstrate
the robustness and reliability of our approach for a series of examples.
