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Error Inhibiting Methods for Finite Elements

Adi Ditkowski, Anne Le Blanc, Chi-Wang Shu

TL;DR

The paper develops Block Finite Difference schemes for the heat equation, showing that BFD can be interpreted as a $p=1$ nodal-based DG method to enable a standard stability analysis and to prove optimal convergence. It demonstrates a 1D two-point, fifth-order stencil with a third-order truncation error that can achieve higher accuracy via post-processing; extends the construction to 2D and 3D through tensor products and provides Dirichlet-boundary adaptations using ghost points. The authors present a unified framework connecting BFD to DG, derive stability proofs, and validate performance with extensive numerical experiments in periodic and Dirichlet settings, including nontrivial boundary treatments and post-processing strategies. The work advances high-order, stable, block-based discretizations for the heat equation and outlines clear paths for applying these methods to more complex geometries and to hyperbolic/advection systems, with potential for significant accuracy and efficiency gains in finite-element/finite-difference hybrids.

Abstract

Finite Difference methods (FD) are one of the oldest and simplest methods for solving partial differential equations (PDE). Block Finite Difference methods (BFD) are FD methods in which the domain is divided into blocks, or cells, containing two or more grid points, with a different scheme used for each grid point, unlike the standard FD method. It was shown in recent works that BFD schemes might be one to three orders more accurate than their truncation errors. Due to these schemes' ability to inhibit the accumulation of truncation errors, these methods were called Error Inhibiting Schemes (EIS). This manuscript shows that our BFD schemes can be viewed as a particular type of Discontinuous Galerkin (DG) method. Then, we prove the BFD scheme's stability using the standard DG procedure while using a Fourier-like analysis to establish its optimal convergence rate. We present numerical examples in one and two dimensions to demonstrate the efficacy of these schemes.

Error Inhibiting Methods for Finite Elements

TL;DR

The paper develops Block Finite Difference schemes for the heat equation, showing that BFD can be interpreted as a nodal-based DG method to enable a standard stability analysis and to prove optimal convergence. It demonstrates a 1D two-point, fifth-order stencil with a third-order truncation error that can achieve higher accuracy via post-processing; extends the construction to 2D and 3D through tensor products and provides Dirichlet-boundary adaptations using ghost points. The authors present a unified framework connecting BFD to DG, derive stability proofs, and validate performance with extensive numerical experiments in periodic and Dirichlet settings, including nontrivial boundary treatments and post-processing strategies. The work advances high-order, stable, block-based discretizations for the heat equation and outlines clear paths for applying these methods to more complex geometries and to hyperbolic/advection systems, with potential for significant accuracy and efficiency gains in finite-element/finite-difference hybrids.

Abstract

Finite Difference methods (FD) are one of the oldest and simplest methods for solving partial differential equations (PDE). Block Finite Difference methods (BFD) are FD methods in which the domain is divided into blocks, or cells, containing two or more grid points, with a different scheme used for each grid point, unlike the standard FD method. It was shown in recent works that BFD schemes might be one to three orders more accurate than their truncation errors. Due to these schemes' ability to inhibit the accumulation of truncation errors, these methods were called Error Inhibiting Schemes (EIS). This manuscript shows that our BFD schemes can be viewed as a particular type of Discontinuous Galerkin (DG) method. Then, we prove the BFD scheme's stability using the standard DG procedure while using a Fourier-like analysis to establish its optimal convergence rate. We present numerical examples in one and two dimensions to demonstrate the efficacy of these schemes.

Paper Structure

This paper contains 25 sections, 85 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Block Finite Difference schemes
  3. Description of Two Points Block, 5th order scheme
  4. Equivalency between the BFD and the DG Methods
  5. The DG Scheme
  6. Generalization to non periodic boundary conditions
  7. Adapting the BFD scheme to non-periodic boundary conditions of Dirichlet type
  8. Equivalence of the BFD and DG schemes
  9. Numerical Results
  10. Conclusion
  11. Generalization to Two Dimensions
  12. Proof of Optimal Convergence
  13. Proof of Stability
  14. Numerical Results
  15. Non-Periodic Boundary Conditions
  16. ...and 10 more sections

Figures (9)

  • Figure 1: Error and Convergence plots, $\log_{10} \| {{\bf E}} \|$ vs. $\log_{10} h$ for 1D Heat Problem - Dirichlet boundary conditions. Left: without Post-Processing; right: with Post-Processing.
  • Figure 2: Grid in 2 Dimensions - Illustration
  • Figure 3: Illustration of the dependence on node ${{\bf x}} _{i-1/2,j-1/2}$ from neighbouring nodes in 2D
  • Figure 4: 2D Heat Problem - Two Points Block, BFD scheme - Periodic BC - Error at Final Time $T=1$ - $N=50$ - No post-processing
  • Figure 5: 2D Heat Problem - Two Points Block, BFD scheme, Convergence plots, $\log_{10} \| {{\bf E}} \|$ vs. $\log_{10} h$ - Periodic BC - Left: no post-processing; right: Spectral post-processing
  • ...and 4 more figures

Theorems & Definitions (1)

  • Definition 1: The Operator ${\Theta}_{j-1/2}$