Blending low-order stabilised finite element methods

a positivity-preserving local projection method for the convection-diffusion equation

Gabriel Barrenechea, Erik Burman, Fotini Karakatsani

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Abstract

In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection-diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.
Original languageEnglish
Pages (from-to)1169-1193
Number of pages25
JournalComputer Methods in Applied Mechanics and Engineering
Volume317
Early online date20 Jan 2017
DOIs
Publication statusPublished - 15 Apr 2017

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convection-diffusion equation
Stabilized Finite Element Method
Convection-diffusion Equation
Projection Method
Positivity
preserving
finite element method
projection
Finite element method
Stabilization
stabilization
maximum principle
A.s. Convergence
Maximum principle
range (extremes)
Smooth Solution
Extremum
Maximum Principle
Monotonicity
Exact Solution

Keywords

  • discrete maximum principle
  • convection-diffusion equation
  • nonlinear scheme
  • local projection stabilisation

Cite this

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AU - Barrenechea, Gabriel

AU - Burman, Erik

AU - Karakatsani, Fotini

PY - 2017/4/15

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N2 - In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection-diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.

AB - In this work we propose a nonlinear blending of two low-order stabilisation mechanisms for the convection-diffusion equation. The motivation for this approach is to preserve monotonicity without sacrificing accuracy for smooth solutions. The approach is to blend a first-order artificial diffusion method, which will be active only in the vicinity of layers and extrema, with an optimal order local projection stabilisation method that will be active on the smooth regions of the solution. We prove existence of discrete solutions, as well as convergence, under appropriate assumptions on the nonlinear terms, and on the exact solution. Numerical examples show that the discrete solution produced by this method remains within the bounds given by the continuous maximum principle, while the layers are not smeared significantly.

KW - discrete maximum principle

KW - convection-diffusion equation

KW - nonlinear scheme

KW - local projection stabilisation

U2 - 10.1016/j.cma.2017.01.016

DO - 10.1016/j.cma.2017.01.016

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JO - Computer Methods in Applied Mechanics end Engineering

JF - Computer Methods in Applied Mechanics end Engineering

SN - 0045-7825

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