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

DOI10.1016/J.CAMWA.2021.10.015arXiv2012.09594MaRDI QIDQ6356335

Brendan Keith

Publication date: 17 December 2020

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 $mathcalLmathcal{L}^{*}$ 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.

Mathematics Subject Classification ID

Numerical optimization and variational techniques (65K10) Boundary value problems for second-order elliptic equations (35J25) Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Rate of convergence, degree of approximation (41A25) PDE constrained optimization (numerical aspects) (49M41)

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