Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2410.22052

Error estimates for perturbed variational inequalities of the first kind

Lothar Banz·Miriam Schönauer·Andreas Schröder
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we derive a priori error estimates for variational inequalities of the first kind in an abstract framework. This is done by combining the first Strang Lemma and the Falk Theorem. The main application consists in the derivation of a priori error estimates for Galerkin methods, in which "variational crimes" may perturb the underlying variational inequality. Different types of perturbations are incorporated into the abstract framework and discussed by various examples. For instance, the perturbation caused by an inexact quadrature is examined in detail for the Laplacian obstacle problem. For this problem, guaranteed rates for the approximation error resulting from the use of higher-order finite elements are derived. In numerical experiments, the influence of the number of quadrature points on the approximation error and on the quadrature-related error itself is studied for several discretization methods.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lothar Banz, Miriam Schönauer, Andreas Schröder. 2024-10-29. Error estimates for perturbed variational inequalities of the first kind. https://arxiv.org/abs/2410.22052

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Fully spectral scheme for the linear BGK equation on the whole space

In this article, we design a fully spectral method in both space and velocity for a linear inhomogeneous kinetic equation with mass, momentum and energy conservation. We focus on the linear BGK equation with a confinement potential $Φ$, even if the method could be applied to different collision operators. It is based upon the projection on Hermite polynomials in velocity and orthonormal polynomials with respect to the weight $e^{-$Φ$}$ in space. The potential $Φ$ is assumed to be a polynomial. It is, to the author's knowledge, the first scheme which preserves hypocoercive behavior in addition to the conservation laws. These different properties are illustrated numerically on both quadratic and double well potential.

math.NA↗
arXiv2026

Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to the full range of lower and upper orders, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.

math.NA↗
arXiv2026

Error Estimates for Hyperbolic Scaling Limits of Linear Kinetic Models on Networks

This paper studies linear discrete kinetic models on networks and their asymptotic behavior in the small Knudsen number limit. For coupling conditions at an n-edge junction under a symmetric formulation, we introduce a change of variables that reformulates the system into n independent initial-boundary value problems. The asymptotic expansions are then constructed and rigorously justified by deriving an error estimate based on the energy method.

math.NA↗