Adaptivity and Error Estimation for the Finite Element Method Applied to Convection Diffusion Problems

A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto "upwinded" polynomials. The err...

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Bibliographic Details
Published in:SIAM journal on numerical analysis Vol. 21; no. 5; pp. 910 - 954
Main Authors: Szymczak, W. G., Babuška, I.
Format: Journal Article
Language:English
Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.10.1984
Subjects:
Approximation
Boundary layers
Convection
Degrees of polynomials
Error rates
Estimates
Estimation methods
Estimators
Exact sciences and technology
Finite element analysis
Finite element method
Greens function
Mathematics
Methods
Numerical analysis
Numerical analysis. Scientific computation
Ordinary differential equations
Polynomials
Sciences and techniques of general use
Non linear equation
Finite element method
Second order equation
Boundary value problem
Error estimation
Differential equation
ISSN:0036-1429, 1095-7170
Online Access:Get full text
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Summary:A detailed analysis is performed for a finite element method applied to the general one-dimensional convection diffusion problem. Piecewise polynomials are used for the trial space. The test space is formed by locally projecting L-spline basis functions onto "upwinded" polynomials. The error is measured in the Lp mesh dependent norm. The method is shown to be quasi-optimal, provided that the input data is piecewise smooth--a reasonable assumption in practice. A posteriori error estimates are derived having the property that the effectivity index θ = (error estimate/true error) converges to one as the maximum mesh size goes to zero. These error estimates are composed of locally computable error indicators, providing for an adaptive mesh refinement strategy. Numerical results show that θ is nearly one even on coarse meshes, and optimal rates of convergence are attained by the adaptive procedure. The robustness of the algorithm is tested on a nonlinear turning point problem modeling flow through an expanding duct.
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ISSN:0036-1429
1095-7170
DOI:10.1137/0721059