A \(C^{1}\)-conforming \(hp\) finite element method for fourth order singularly perturbed boundary value problems

DOI10.1016/J.APNUM.2016.02.002zbMath1336.65124OpenAlexW2260148615MaRDI QIDQ268848

Pandelitsa Panaseti, Niall Madden, Antri Zouvani, Christos Xenophontos

Publication date: 15 April 2016

Published in: Applied Numerical Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.apnum.2016.02.002

zbMATH Keywords

numerical exampleboundary layersexponential convergence\(hp\) finite element methodfourth-order singularly perturbed boundary value problemspectral boundary layer mesh

Mathematics Subject Classification ID

Stability and convergence of numerical methods for ordinary differential equations (65L20) Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations (65L60) Linear boundary value problems for ordinary differential equations (34B05) Singular perturbations for ordinary differential equations (34E15) Mesh generation, refinement, and adaptive methods for ordinary differential equations (65L50) Numerical solution of singularly perturbed problems involving ordinary differential equations (65L11)

Related Items (13)

On the finite element approximation of fourth-order singularly perturbed eigenvalue problems ⋮ An \(hp\) finite element method for \(4^{\mathrm{th}}\) order singularly perturbed problems ⋮ An \(hp\) finite element method for a 4th order singularly perturbed boundary value problem in two dimensions ⋮ Solving a singular beam equation by the method of energy boundary functions ⋮ The Bogner-Fox-Schmit element finite volume methods on the Shishkin mesh for fourth-order singularly perturbed elliptic problems ⋮ Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE ⋮ A $C^1$ Conforming Petrov--Galerkin Method for Convection-Diffusion Equations and Superconvergence Analysis over Rectangular Meshes ⋮ A parameter robust finite element method for fourth order singularly perturbed problems ⋮ Convergence analysis of the Adini element on a Shishkin mesh for a singularly perturbed fourth-order problem in two dimensions ⋮ Robust error estimation in energy and balanced norms for singularly perturbed fourth order problems ⋮ Computational approach for fourth-order self-adjoint singularly perturbed boundary value problems via non-polynomial quintic spline ⋮ A mixed hp FEM for the approximation of fourth‐order singularly perturbed problems on smooth domains ⋮ Numerical treatment of singularly perturbed fourth-order two-parameter problems

Cites Work

• Robust exponential convergence of \(hp\)-FEM in balanced norms for singularly perturbed reaction-diffusion equations
• Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales
• \(hp\)-finite element methods for singular perturbations
• Error estimation in a balanced norm for a convection-diffusion problem with two different boundary layers
• On the Finite Element Approximation of Systems of Singularly Perturbed Rection-diffusion Equations
• Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction‐diffusion problems
• Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I: reaction-diffusion-type problems
• On the robust exponential convergence of hp finite element methods for problems with boundary layers
• The $p$ and $hp$ versions of the finite element method for problems with boundary layers
• Robust exponential convergence of hp FEM for singularly perturbed reaction-diffusion systems with multiple scales
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