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Consider the singularly perturbed linear reaction-diffusion problem −ε 2 Δu + bu = f in Ω ⊂ R d , u = 0 on ∂Ω, where d ≥ 1, the domain Ω is bounded with (when d ≥ 2) Lipschitz-continuous boundary ∂Ω, and the parameter ε satisfies 0 < ε 1. It is argued that for this type of problem, the standard energy norm v → [ε 2 |v| 2 1 + v 2 0 ] 1/2 is too weak a norm… (More)
In this work, we analytically identify natural superconvergent points of function values and gradients for triangular elements. Both the Poisson equation and the Laplace equation are discussed for polynomial finite element spaces (with degrees up to 8) under four different mesh patterns. Our results verify computer findings of , especially, we confirm… (More)
The ultraconvergence property of the Zienkiewicz-Zhu gradient patch recovery technique based on local discrete least-squares fitting is established for a large class of even-order finite elements. The result is valid at all rectangular mesh symmetry points. Different smoothing strategies are discussed and numerical examples are demonstrated.
A systematic and analytic process is conducted to identify natural superconvergence points of high degree polynomial C 0 finite elements in a three-dimensional setting. This identification is based upon explicitly constructing an orthogonal decomposition of local finite element spaces. Derivative and function value superconvergence points are investigated… (More)
In , we analytically identified natural superconvergent points of function values and gradients for several popular three-dimensional polynomial finite elements via an orthogonal decomposition. This paper focuses on the detailed process for determining the superconvergent points of pentahedral and tetrahedral elements.
A system of linear coupled reaction-diffusion equations is considered, where each equation is a two-point boundary value problem and all equations share the same small diffusion coefficient. A finite element method using piecewise quadratic splines that are globally C 1 is introduced; its novelty lies in the norm associated with the method, which is… (More)