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Results on a Mixed Finite Element Approach for a Model Convection-Diffusion Problem

Constantin Bacuta, Daniel Hayes, Tyler O'Grady

TL;DR

The paper tackles a singularly perturbed convection-diffusion model in the convection-dominated regime and develops a unified stability framework via optimal norms for several finite element discretizations. It compares standard linear Galerkin, SPLS with a quadratic test space, and upwind PG/SD schemes within a mixed formulation, uncovering how residual-based SPLS and bubble-enriched test spaces mitigate non-physical oscillations while preserving accuracy. A key result is that upwind PG, viewed through the SPLS lens, delivers strong stability and potential $\mathcal{O}(h^2)$ convergence in the appropriate norm for $\varepsilon<h$, outperforming SD in many norms. The methods and analysis lay groundwork for efficient, stable discretizations of more general multidimensional singular perturbation problems with convection dominance.

Abstract

We consider a model convection-diffusion problem and present our recent numerical and analysis results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models

Results on a Mixed Finite Element Approach for a Model Convection-Diffusion Problem

TL;DR

The paper tackles a singularly perturbed convection-diffusion model in the convection-dominated regime and develops a unified stability framework via optimal norms for several finite element discretizations. It compares standard linear Galerkin, SPLS with a quadratic test space, and upwind PG/SD schemes within a mixed formulation, uncovering how residual-based SPLS and bubble-enriched test spaces mitigate non-physical oscillations while preserving accuracy. A key result is that upwind PG, viewed through the SPLS lens, delivers strong stability and potential convergence in the appropriate norm for , outperforming SD in many norms. The methods and analysis lay groundwork for efficient, stable discretizations of more general multidimensional singular perturbation problems with convection dominance.

Abstract

We consider a model convection-diffusion problem and present our recent numerical and analysis results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models
Paper Structure (20 sections, 9 theorems, 111 equations, 1 figure, 5 tables)

This paper contains 20 sections, 9 theorems, 111 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. The notation and the general SPLS approach
  3. The abstract variational formulation at the continuous level
  4. Saddle point least squares discretization
  5. Discretization with $C^0-P^1$ trial space for the 1D Convection reaction problem
  6. Standard Linear discretization
  7. SPLS discretization
  8. Petrov Galerkin (PG) with bubble enriched test space $V_h$
  9. Upwinding Petrov Galerkin discretization
  10. Variational formulation and matrices
  11. Stream line diffusion (SD) discretization
  12. Stability of mixed discretization for the 1D Convection reaction problem
  13. Optimal trial norms
  14. Analysis of the standard linear discretization with optimal trial norm
  15. Analysis of the SPLS with quadratic test space $V_h$
  16. ...and 5 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 unique solution that depends continuously on the data $F$.

Figures (1)

  • Figure 6.1: $\varepsilon = 10^{-6}$. Left: Linear, Right: SPLS

Theorems & Definitions (14)

  • Proposition 2.1
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Proposition 5.1
  • proof
  • Theorem 5.2
  • Theorem 5.3
  • Theorem 5.4
  • proof
  • ...and 4 more