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

• Published in: Numerical methods for partial differential equations 2017-11, Vol.33 (6), p.2005-2022
• Main Authors: Amrein, Mario, Wihler, Thomas P.
• Format: Article
• Language: English
• Subjects: adaptive finite element methods; adaptive pseudo transient continuation method; Boundary element method; Boundary value problems; Differential equations; dynamical system; Finite element method; Galerkin method; Mathematical analysis; Partial differential equations; Robustness (mathematics); semilinear elliptic problems; singularly perturbed problems; steady states
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.22177

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.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 2005–2022, 2017