Error inhibiting methods for finite elements

¹ School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
² Department of Geophysics, Porter school of Environment and Earth Sciences, Tel Aviv University, Tel Aviv 69978, Israel
³ Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

^* Corresponding author: anneleb@tauex.tau.ac.il

Received: 12 May 2024
Accepted: 16 December 2024

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.