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Comparison of variational discretizations for a convection-diffusion problem

Constantin Bacuta, Cristina Bacuta, Daniel Hayes

Abstract

For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is based on finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare the standard linear discretization, the saddle point least square and upwinding Petrov-Galerkin methods. We conclude that the bubble upwinding Petrov-Galerkin method is the most performant discretization for the one dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multidimensional cases.

Comparison of variational discretizations for a convection-diffusion problem

Abstract

For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is based on finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare the standard linear discretization, the saddle point least square and upwinding Petrov-Galerkin methods. We conclude that the bubble upwinding Petrov-Galerkin method is the most performant discretization for the one dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multidimensional cases.
Paper Structure (17 sections, 13 theorems, 121 equations)

This paper contains 17 sections, 13 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. The general mixed variational approach
  3. The abstract variational formulation at the continuous level
  4. PG and SPLS discretizations
  5. Optimal trial norm for the convection diffusion problem
  6. Discrete optimal trial norm
  7. Discrete optimal trial norm for the one dimensional case
  8. Standard and and SPLS finite element variational formulation and discretization
  9. Standard discretization with $C^0-P^1$ test and trial spaces
  10. SPLS discretization
  11. The oscillatory behavior of the $P^1-P^2$ SPLS discretization
  12. The Petrov-Galerkin method with bubble type test space
  13. Upwinding PG with particular bubble functions
  14. Upwinding PG with quadratic bubble functions
  15. Special cases for quadratic bubble upwinding
  16. ...and 2 more sections

Key Result

Proposition 2.1

If the form $b(\cdot,\cdot)$ satisfies sup-sup_a and inf-sup_a, and the data $F \in V^*$ satisfies the compatibility conditioneq:BBsuf, then the problem VFabstract has a unique solution that depends continuously on the data $F$.

Theorems & Definitions (19)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • proof
  • ...and 9 more