Error Inhibiting Methods for Finite Elements

TL;DR

This paper introduces Error Inhibiting Schemes (EIS) within Block Finite Difference methods, demonstrating their stability, optimal convergence, and relation to Discontinuous Galerkin methods through theoretical analysis and numerical experiments.

Contribution

It shows that BFD schemes can be viewed as a type of DG method, proves their stability, and establishes their optimal convergence rate.

Findings

BFD schemes can be interpreted as a specific type of DG method.

The schemes are proven to be stable using standard DG procedures.

Numerical examples confirm the schemes' effectiveness in 1D and 2D.

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.