The defect correction principle and discretization methods

Stetter, Hans J.

doi:10.1007/bf01432879

Cited by 246 publications

(136 citation statements)

References 14 publications

(22 reference statements)

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“…The iterative defect-correction method (see e.g. [37]) is an iterative procedure which aims at increasing the accuracy of a numerical solution, without mesh refinement. More precisely, to approximate the solution x * of the equation…”

Section: Improving Time Accuracy: Defect-correction Iterationsmentioning

confidence: 99%

“…In order to recover optimal accuracy, we propose an improved explicit coupling scheme involving a few defect-correction iterations (see e.g. [37]). Numerical experiments, in the linear and non-linear case, show that optimal time accuracy can be obtained by performing one defect-correction iteration (when first order accuracy is expected for the underlying implicit coupling scheme).…”

Section: Introductionmentioning

confidence: 99%

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Stabilization of explicit coupling in fluid–structure interaction involving fluid incompressibility

Burman

Fernández

2009

Computer Methods in Applied Mechanics and Engineering

166

208

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In this work we propose a stabilized explicit coupling scheme for fluid-structure interaction problems involving a viscous incompressible fluid. The coupled discrete formulation is based on Nitsche's method with a time penalty term giving L 2 -control on the fluid pressure variations at the interface. The scheme is stable, in the energy norm, irrespectively of the fluid-structure density ratio. Numerical experiments, in two and three dimensions, show that optimal time accuracy can be obtained by performing a few defect-correction iterations.Key-words: Fluid-structure interaction, Nitsche's method, fluid incompressibility, time discretization, explicit coupling, loosely coupled schemes, defectcorrection method

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Section: Improving Time Accuracy: Defect-correction Iterationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stabilization of explicit coupling in fluid–structure interaction involving fluid incompressibility

Burman

Fernández

2009

Computer Methods in Applied Mechanics and Engineering

166

208

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“…If its accuracy is less, we can iterate with the lumped mass matrix as preconditioner. This approach resembles defect correction (Stetter, 1978), which has the following convenient property. Consider two operators L 1 and L 2 where L k has an order of accuracy p k (k = 1, 2) and p 1 > p 2 .…”

Section: Methodsmentioning

confidence: 99%

Dispersion Analysis of Finite-element Schemes for a First-order Formulation of the Wave Equation

Shamasundar¹,

Al‐Khoury²,

Mulder³

2015

Proceedings

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SUMMARYWe investigated one-dimensional numerical dispersion curves and error behaviour of four finite-element schemes with polynomial basis functions: the standard elements with equidistant nodes, the LegendreGauss-Lobatto points, the Chebyshev-Gauss-Lobatto nodes without a weighting function and with. Mass lumping, required for efficiency reasons and enabling explicit time stepping, may adversely affect the numerical error. We show that in some cases, the accuracy can be improved by applying one iteration on the full mass matrix, preconditioned by its lumped version. For polynomials of degree one, this improves the accuracy from second to fourth order in the element size. In other cases, the improvement in accuracy is less dramatic.

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“…These techniques are a form of non-iterative difference or differential defect correction [1,9]. In all cases in the error transport literature, the error equations are linearized, and the focus is typically on the approximation used to evaluate the local residual, that is, the local source (sink) of error that drives an error transport equation.…”

Section: Introductionmentioning

confidence: 99%

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

Banks

Hittinger

Connors

et al. 2012

Computer Methods in Applied Mechanics and Engineering

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The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to solve an auxiliary error equation concurrently with the primal governing equation to obtain a point-wise (cell-wise) estimate of the discretization error. One-dimensional, nonlinear, time-dependent problems are considered. In contrast to previous work, fully nonlinear error equations are advanced, and potential benefits are identified. A systematic approach to approximate the local residual for both method-of-lines and space-time discretizations is developed. Behavior of the error estimates on problems that include weak solutions demonstrates that this nonlinear error evolution provides useful error estimates.

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The defect correction principle and discretization methods

Cited by 246 publications

References 14 publications

Stabilization of explicit coupling in fluid–structure interaction involving fluid incompressibility

Stabilization of explicit coupling in fluid–structure interaction involving fluid incompressibility

Dispersion Analysis of Finite-element Schemes for a First-order Formulation of the Wave Equation

Numerical error estimation for nonlinear hyperbolic PDEs via nonlinear error transport

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