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In this paper we investigate the application of pseudo-transient-continuation(PTC) schemes for the numerical solution of semilinear elliptic partialdifferential equations, with possible singular perturbations. We will outline aresidual reduction analysis within the framework of general Hilbert spaces,and, subsequently, employ the PTC-methodology in the context of finite elementdiscretizations of semilinear boundary value problems. Our approach combinesboth a prediction-type PTC-method (for infinite dimensional problems) and anadaptive finite element discretization (based on a robust a posteriori residualanalysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numericalexperiments underline the robustness and reliability of the proposed approachfor different examples.

Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations