Element-based B-spline basis function spaces: construction and application in isogeometric analysis

Abstract: This paper develops a unified theoretical framework for constructing B-spline basis function spaces with structural equivalence to finite element spaces. The theory rigorously establishes that these bases emerge as explicit linear combinations of B-spline element bases. For any prescribed smoothness requirements, this element-wise formulation enables the Hermite interpolation at nodes, which directly utilizes function values and derivatives without solving global linear systems. By focusing on explicit interpolation properties, element-wise analysis establishes optimal approximation errors, even when the space smoothness attains its theoretical maximum for the space degree. In isogeometric analysis (IgA), the construction naturally decomposes geometric mappings into element-level representations, allowing efficient computations across elements regardless of node distribution. Notably, the same Hermite interpolation framework simultaneously handles domain parameterization and IgA solutions, allowing direct imposition of boundary conditions through function and derivative matching. Numerical tests demonstrate optimal convergence rates and superconvergence properties in 2D IgA under uniform knot configurations, and improved computational efficiency in 3D IgA with non-uniform knot distributions.