Adaptive pseudo-transient-continuation-Galerkin methods for semilinear elliptic partial differential equations

Amrein, Mario; Wihler, Thomas (2017). Adaptive pseudo-transient-continuation-Galerkin methods for semilinear elliptic partial differential equations. Numerical methods for partial differential equations, 33(6), pp. 2005-2022. Wiley 10.1002/num.22177

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In this article, we investigate the application of pseudo-transient-continuation (PTC) schemes for the numerical solution of semilinear elliptic partial differential equations, with possible singular perturbations. We will outline a residual reduction analysis within the framework of general Hilbert spaces, and, subsequently, use the PTC-methodology in the context of finite element discretizations of semilinear boundary value problems. Our approach combines both a prediction-type PTC-method (for infinite dimensional problems) and an adaptive finite element discretization (based on a robust a posteriori residual analysis), thereby leading to a fully adaptive PTC -Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for different examples

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