An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

Houston, Paul; Wihler, Thomas P.

doi:10.1090/mcom/3308

Cited by 18 publications

(20 citation statements)

References 56 publications

(101 reference statements)

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“…The following result, which implies the well-posedness of (8) even in the incompressible limit ν = 1 /2, follows immediately from [24, Theorem 5.1]. (13) holds true, then we have the inf-sup condition (14) γ a := inf…”

Section: Faces and Face Operatorsmentioning

confidence: 85%

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

Wihler

Wirz

2020

J Sci Comput

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We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in R 3 . In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.

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Section: Faces and Face Operatorsmentioning

confidence: 85%

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

Wihler

Wirz

2020

J Sci Comput

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“…Such a setting has been considered in, e.g., [12,19], where reliable (guaranteed) and efficient a posteriori error estimates were derived. Adaptive algorithms aiming at a balance of the linearization and discretization errors were proposed and their optimal performance was observed numerically; see, e.g., [1,4,13,28]. Later, theoretical proofs of plain convergence (without rates) were given in [25,30], where [30] builds on the unified framework of [29] encompassing also the Kačanov and (damped) Newton linearizations in addition to the Banach-Picard linearization (6).…”

Section: Taking Into Account the Discretization And Linearization Errorsmentioning

confidence: 99%

Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

et al. 2021

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We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the Banach–Picard linearization, and a contractive linear algebraic solver. In particular, we identify stopping criteria for the algebraic solver that on the one hand do not request an overly tight tolerance but on the other hand are sufficient for the inexact (perturbed) Banach–Picard linearization to remain contractive. Similarly, we identify suitable stopping criteria for the Banach–Picard iteration that leave an amount of linearization error that is not harmful for the residual a posteriori error estimate to steer reliably the adaptive mesh-refinement. For the resulting algorithm, we prove a contraction of the (doubly) inexact iterates after some amount of steps of mesh-refinement/linearization/algebraic solver, leading to its linear convergence. Moreover, for usual mesh-refinement rules, we also prove that the overall error decays at the optimal rate with respect to the number of elements (degrees of freedom) added with respect to the initial mesh. Finally, we prove that our fully adaptive algorithm drives the overall error down with the same optimal rate also with respect to the overall algorithmic cost expressed as the cumulated sum of the number of mesh elements over all mesh-refinement, linearization, and algebraic solver steps. Numerical experiments support these theoretical findings and illustrate the optimal overall algorithmic cost of the fully adaptive algorithm on several test cases.

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“…We remark that, although (A.1) and (A.2) are not necessarily satisfied for this problem, our fully adaptive PTC‐Galerkin approach still delivers accurate results. We note, however, that divergence and/or chaotic behavior of the iterative procedure in Algorithm 1 may occur if initial guesses

u_{0}^{h}

are too far away from possible solutions of the PDE (see also , , ]).…”

Section: Application To Semilinear Problemsmentioning

confidence: 99%

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

Amrein

Wihler

2017

Numerical Methods Partial

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Abstract. In this paper 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, employ 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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An $hp$-adaptive Newton-discontinuous-Galerkin finite element approach for semilinear elliptic boundary value problems

Cited by 18 publications

References 56 publications

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

Convergence and quasi-optimal cost of adaptive algorithms for nonlinear operators including iterative linearization and algebraic solver

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

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