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.

翻译：在基于非插值函数（即不具备克罗内克δ性质的函数）的伽辽金型方法中，施加非齐次狄利克雷（本质）边界条件是一项基本挑战。此类函数通常用于各种无网格方法以及基于等几何范式的数值方法。本文分析了一个由非标准边界条件下的泊松方程构成的模型问题。具体而言，该模型问题不采用经典边界条件，而是涉及狄利克雷型和非齐次纽曼型非局部边界条件。本文推导了强形式和弱形式施加非齐次狄利克雷型非局部条件的变分公式，并在基于非均匀有理B样条的等几何框架内，通过广泛的数值研究对二者进行了比较。数值研究的重点在于非局部边界条件对所考虑离散化方法性质的影响。