A Recovery-Based Error Indicator for Finite Difference Methods

TL;DR

This work introduces a recovery-based a posteriori error indicator for high-order finite difference methods by interpolating the FD solution into a polynomial finite element space and applying Polynomial Preserving Recovery to estimate gradient errors. The method is equation-agnostic and delivers both local and global indicators with theoretical guarantees, including consistency, boundedness, and a superconvergence-based asymptotic exactness of the estimator in the gradient norm. Through Poisson and wave-equation experiments in 2D and 3D (including discontinuous coefficients and heterogeneous media), the authors demonstrate that the indicator accurately reflects the true gradient error in the RMS natural norm and captures local error features, with spatial indicators typically performing better than temporal ones. The results support using this recovery-based estimator as a robust, black-box tool for error evaluation and potential adaptivity in complex multiphysics simulations.

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.

Figures (18)

• Figure 1: Triangular mesh. The local patches $\mathcal{K}_i,\; \mathcal{K}_j,$ and $\mathcal{K}_k$ for respectively the boundary vertices $z_i$ and $z_j$ and the internal vertex $z_k$ are shown in green. On the left we have the the case $r = 1$ and $d = 2$ and on the right $r = 2$ and $d = 2$
• Figure 1: Interpolant mesh (blue solid lines) and FD grid (gray dotted lines) for $N = 5$ and $r = 3$ together with the one-dimensional cubic Lagrange basis functions in each spatial direction
• Figure 1: The gradient of the error, the indicator, the error of the recovered gradient, for the poisson problem with exact solution $u(x_1,x_2) = \sin{8\pi x_1}\sin{8\pi x_2}$ and various values of $h$, defined through $N+1$. For $r = 1$ on the left, $r = 3$ on the right and $N+1 = 6, 12, 24, 48, 96$ and $192$.
• Figure 2: Quadrilateral mesh. The local patches $\mathcal{K}_i,\; \mathcal{K}_j,$ and $\mathcal{K}_k$ for respectively the boundary vertices $z_i$ and $z_j$ and the internal vertex $z_k$ are shown in green. On the left we have the the case $r = 1$ and $d = 2$ and on the right $r = 2$ and $d = 2$
• Figure 2: Local results for $r = 1$ and $h = \frac{1}{48}$ for the poisson problem with exact solution $u(x_1,x_2) = \sin{8\pi x_1}\sin{8\pi x_2}$. On the left is the local error, and on the right is the local indicator. Additionally, we used the same color scale in both plots.

Theorems & Definitions (4)

• Proposition 3.1
• Proof 1
• Proposition 3.2
• Proof 2