Optimal Approximation in Sobolev Spaces: A New Approach of Spline Functions

In this article, the interaction of spline functions with Sobolev spaces in the numericalsolution of partial differential equations (PDEs) is examined from a new andcomprehensive perspective. Sobolev spaces, thanks to their integrability of derivativesand suitable norm structures, provide a powerful framework for the solutiontheory of PDEs. In recent years, spline-based approaches, which have emerged asalternatives to classical finite element methods (FEM), have attracted attentionparticularly due to their advantages such as high-order derivative continuity andadaptive knot selection. This approach can produce effective and accurate solutionsnot only in physical applications such as fluid mechanics or elasticity problems butalso in a wide range including heat transfer, biological modeling, and financialderivatives pricing.The main novelty of this article is to systematically examine the optimal approximationproperties of spline functions in Sobolev norms in the light of embeddingtheorems. In this way, it becomes clearer how critical issues such as the compatibilityof piecewise polynomials with boundary conditions and derivative continuityare in terms of numerical stability and solution accuracy. Moreover, when combinedwith the isogeometric analysis (IGA) approach, it is shown that spline-basedfunctions can also work smoothly on geometric definitions directly obtained fromengineering design data (e.g., CAD models). Thus, a method emerges that bothreduces computational cost and ensures high accuracy.This study also details the underlying mathematical principles of the optimalapproximation provided by spline functions in Sobolev spaces; the connection betweentheory and application is supported by numerical experiments on sample PDEproblems. The results obtained reveal that, compared to classical approaches, thesame or better accuracy can be achieved with fewer degrees of freedom. In this way,it provides significant motivation for further development of spline-based methodsin both theoretical and computational aspects for future research. As a result, thisarticle aims to serve as an important guide for obtaining highly accurate and efficientsolutions by offering new insights into the interaction of Sobolev spaces andspline functions in solving partial differential equations.