Imposing nonlocal boundary conditions in Galerkin-type methods based on non-interpolatory functions

TL;DR

This paper investigates how to impose nonlocal boundary conditions in Galerkin methods using non-interpolatory functions, focusing on variational formulations and their effects in isogeometric analysis with NURBS.

Contribution

It introduces and compares approaches for imposing nonlocal boundary conditions in Galerkin methods based on non-interpolatory functions, especially within isogeometric analysis.

Findings

Strongly imposed boundary conditions influence discretisation properties.

Weak imposition offers different numerical stability characteristics.

Nonlocal boundary conditions significantly affect solution accuracy.

Abstract

The imposition of inhomogeneous Dirichlet (essential) boundary conditions is a fundamental challenge in the application of Galerkin-type methods based on non-interpolatory functions, i.e., functions which do not possess the Kronecker delta property. Such functions typically are used in various meshfree methods, as well as methods based on the isogeometric paradigm. The present paper analyses a model problem consisting of the Poisson equation subject to non-standard boundary conditions. Namely, instead of classical boundary conditions, the model problem involves Dirichlet- and Neumann-type nonlocal boundary conditions. Variational formulations with strongly and weakly imposed inhomogeneous Dirichlet-type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non-uniform rational B-splines (NURBS). The attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods.