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Connections Between Finite Difference and Finite Element Approximations for a Convection-Diffusion Problem

Constantin Bacuta, Cristina Bacuta

TL;DR

The paper builds a rigorous bridge between finite difference and finite element discretizations for a 1D convection-diffusion problem in the convection-dominated regime by introducing bubble-type test spaces in a Petrov-Galerkin FE framework. It demonstrates that, with appropriate bubble choice and quadrature, the FE and FD systems share the same stiffness matrix, establishing a direct FD-FE correspondence and enabling improved upwind schemes and new error estimates. A key result is that an exponential bubble PG method can recover the exact nodal interpolant of the true solution, and under suitable quadrature, yields provable convergence bounds for the FD approximation. The methods generalize to higher-order bubble functions, connect to well-known upwind schemes such as IAS/SG, and offer a pathway to efficient discretizations for multidimensional convection-dominated problems.

Abstract

We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov-Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right hand side load vectors for the two methods. This new approach allows for improving well known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov-Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.

Connections Between Finite Difference and Finite Element Approximations for a Convection-Diffusion Problem

TL;DR

The paper builds a rigorous bridge between finite difference and finite element discretizations for a 1D convection-diffusion problem in the convection-dominated regime by introducing bubble-type test spaces in a Petrov-Galerkin FE framework. It demonstrates that, with appropriate bubble choice and quadrature, the FE and FD systems share the same stiffness matrix, establishing a direct FD-FE correspondence and enabling improved upwind schemes and new error estimates. A key result is that an exponential bubble PG method can recover the exact nodal interpolant of the true solution, and under suitable quadrature, yields provable convergence bounds for the FD approximation. The methods generalize to higher-order bubble functions, connect to well-known upwind schemes such as IAS/SG, and offer a pathway to efficient discretizations for multidimensional convection-dominated problems.

Abstract

We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov-Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right hand side load vectors for the two methods. This new approach allows for improving well known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov-Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.
Paper Structure (10 sections, 3 theorems, 112 equations)

This paper contains 10 sections, 3 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Standard Finite Difference Discretization
  3. Finite element linear variational formulation and discrete optimal trial norm
  4. Standard discretization with $C^0-P^1$ test and trial spaces
  5. Optimal discrete trial norms
  6. The Petrov-Galerkin method with bubble type test space
  7. General Bubble Upwinding Petrov-Galerkin Method
  8. Comparing the upwinding FD and the PG FE methods
  9. Upwinding PG with quadratic bubble functions
  10. Upwinding PG with exponential bubble functions

Key Result

Theorem 6.1

Let $u_h := \sum_{i=1}^{n-1} u_i \varphi_i$ be the finite element solution of eq:1d-modelPG with the test space as defined in eq:VhE. Then $u_h$ coincides with the linear interpolant $I_h(u)$ of the exact solution $u$ of eq:1d-model, with $\kappa=1$ on the nodes $x_0,x_1, \ldots, x_n$. In other word

Theorems & Definitions (9)

  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Remark 4.4
  • Theorem 6.1
  • proof
  • Theorem 6.2
  • Corollary 6.3