A priori error analysis of high-order LL* (FOSLL*) finite element methods

TL;DR

This paper provides an a priori error analysis for high-order LL* (FOSLL*) finite element methods, revealing limitations in convergence rates due to the regularity of an auxiliary Lagrange multiplier influenced by domain geometry.

Contribution

It offers the first theoretical analysis explaining why high-order LL* methods have suboptimal convergence rates, highlighting the role of the Lagrange multiplier's regularity.

Findings

Convergence rates are limited by the regularity of the Lagrange multiplier.

High-order methods do not necessarily improve convergence for constant solutions.

Numerical experiments confirm the theoretical predictions.

Abstract

A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are $\mathcal{L}\mathcal{L}^*$ methods. In this work, we argue that all high-order methods in this class should be expected to deliver substandard uniform h-refinement convergence rates. In fact, one may not even see rates proportional to the polynomial order $p > 1$ when the exact solution is a constant function. We show that the convergence rate is limited by the regularity of an extraneous Lagrange multiplier variable which naturally appears via a saddle-point analysis. In turn, limited convergence rates appear because the regularity of this Lagrange multiplier is determined, in part, by the geometry of the domain. Numerical experiments support our conclusions.