Discrete Energy Laws for the First-Order System Least-Squares Finite-Element Approach

TL;DR

This paper investigates how well first-order system least-squares finite-element methods preserve the physical energy laws in time-dependent PDEs, showing they do so with higher accuracy than standard convergence rates.

Contribution

It provides a detailed analysis of discrete energy law adherence in FOSLS methods, including bounds on convergence and an abstract framework for general discretizations.

Findings

Discrete energy laws hold with order O(h^{2p})

Energy law conformance exceeds standard finite-element convergence rates

Numerical experiments confirm theoretical bounds

Abstract

This paper analyzes the discrete energy laws associated with first-order system least-squares (FOSLS) discretizations of time-dependent partial differential equations. Using the heat equation and the time-dependent Stokes' equation as examples, we discuss how accurately a FOSLS finite-element formulation adheres to the underlying energy law associated with the physical system. Using regularity arguments involving the initial condition of the system, we are able to give bounds on the convergence of the discrete energy law to its expected value (zero in the examples presented here). Numerical experiments are performed, showing that the discrete energy laws hold with order $\mathcal O\left(h^{2p}\right)$, where $h$ is the mesh spacing and $p$ is the order of the finite-element space. Thus, the energy law conformance is held with a higher order than the expected, $\mathcal{O}\left(h^p\right)$, convergence of the finite-element approximation. Finally, we introduce an abstract framework for analyzing the energy laws of general FOSLS discretizations.