A practical algorithm to minimize the overall error in FEM computations

TL;DR

This paper introduces a practical algorithm that uses numerical experiments to estimate round-off error growth in FEM, enabling accurate prediction of optimal DoFs and minimal error, significantly reducing computational time.

Contribution

The paper presents a novel method to determine error growth coefficients in FEM, improving accuracy predictions and computational efficiency for general PDE problems.

Findings

Error proportional to N^β_R with estimated coefficients

Strategy predicts minimal achievable error E_min accurately

CPU time reduced by 60-90% for high-accuracy solutions

Abstract

Using the standard finite element method (FEM) to solve general partial differential equations, the round-off error is found to be proportional to $N^{\beta_{\rm R}}$, with $N$ the number of degrees of freedom (DoFs) and $\beta_{\rm R}$ a coefficient. A method which uses a few cheap numerical experiments is proposed to determine the coefficient of proportionality and $\beta_{\rm R}$ in various space dimensions and FEM packages. Using the coefficients obtained above, the strategy put forward in \cite{liu386balancing} for predicting the highest achievable accuracy $E_{\rm min}$ and the associated optimal number of DoFs $N_{\rm opt}$ for specific problems is extended to general problems. This strategy allows predicting $E_{\rm min}$ accurately for general problems, with the CPU time for obtaining the solution with the highest accuracy $E_{\rm min}$ typically reduced by 60\%--90\%.