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Adaptive pseudo‐transient‐continuation‐Galerkin methods for semilinear elliptic partial differential equations
Author(s) -
Amrein Mario,
Wihler Thomas P.
Publication year - 2017
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22177
Subject(s) - mathematics , galerkin method , partial differential equation , finite element method , discretization , robustness (evolution) , continuation , a priori and a posteriori , discontinuous galerkin method , boundary value problem , residual , elliptic partial differential equation , partial derivative , mathematical analysis , computer science , algorithm , biochemistry , chemistry , physics , philosophy , epistemology , gene , thermodynamics , programming language
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