Implicit-Explicit Error Indicator based on Approximation Order

TL;DR

This paper introduces a novel error indicator for PDE solutions that compares implicit and explicit approximations of different orders, aiding adaptive methods by identifying high-error regions.

Contribution

It proposes a new error estimation method based on implicit-explicit approximation order differences, enhancing adaptive PDE solution techniques.

Findings

Effective in locating high-error regions in synthetic tests

Potential for improving adaptive refinement strategies

Demonstrated on a 2D Poisson problem

Abstract

With the immense computing power at our disposal, the numerical solution of partial differential equations (PDEs) is becoming a day-to-day task for modern computational scientists. However, the complexity of real-life problems is such that tractable solutions do not exist. This makes it difficult to validate the numerically obtained solution, so good error estimation is crucial in such cases. It allows the user to identify problematic areas in the computational domain that may affect the stability and accuracy of the numerical method. Such areas can then be remedied by either \textit{h}- or \textit{p}-adaptive procedures. In this paper, we propose to estimate the error of the numerical solution by solving the same governing problem implicitly and explicitly, using a different approximation order in each case. We demonstrate the newly proposed error indicator on the solution of a synthetic two-dimensional Poisson problem with tractable solution for easier validation. We show that the proposed error indicator has good potential for locating areas of high error.