A Recovery-Based Error Indicator for Finite Difference Methods

TL;DR

This paper introduces a recovery-based error indicator for high-order finite difference methods that improves gradient error estimation through polynomial interpolation and is validated across various PDE problems.

Contribution

It presents a novel recovery-based error indicator utilizing polynomial interpolation for finite difference methods, enhancing error estimation accuracy.

Findings

Demonstrates improved gradient error estimation in 2D Poisson problems.

Validates effectiveness on elliptic problems with discontinuous coefficients.

Shows applicability to wave equations in homogeneous and heterogeneous media.

Abstract

A novel recovery-based error indicator for high-order Finite Difference Methods, based on post-processing of the Finite Difference values is presented. The values obtained on the Finite Difference grid are interpolated into a suitable polynomial Finite Element space. A recovery-based error indicator, with the polynomial-preserving property, is then applied to estimate the gradient error. The performance and accuracy of the proposed error indicator are demonstrated through several numerical experiments, including the two-dimensional Poisson problem solved using second- and fourth-order finite difference schemes. Additional experiments are conducted on elliptic problems with discontinuous coefficients, as well as on the two and three-dimensional wave equation in homogeneous media with second- and fourth-order finite differences, and in heterogeneous media with second-order finite differences.