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

In this article, the interaction of spline functions with Sobolev spaces in the numerical solution of partial differential equations (PDEs) is examined from a new and comprehensive perspective. Sobolev spaces, thanks to their integrability of derivatives and suitable norm structures, provide a powerful framework for the solution theory of PDEs. In recent years, spline-based approaches, which have emerged as alternatives to classical finite element methods (FEM), have attracted attention particularly due to their advantages such as high-order derivative continuity and adaptive knot selection. This approach can produce effective and accurate solutions not only in physical applications such as fluid mechanics or elasticity problems but also in a wide range including heat transfer, biological modeling, and financial derivatives pricing. The main novelty of this article is to systematically examine the optimal approximation properties of spline functions in Sobolev norms in the light of embedding theorems. In this way, it becomes clearer how critical issues such as the compatibility of piecewise polynomials with boundary conditions and derivative continuity are in terms of numerical stability and solution accuracy. Moreover, when combined with the isogeometric analysis (IGA) approach, it is shown that spline-based functions can also work smoothly on geometric definitions directly obtained from engineering design data (e.g., CAD models). Thus, a method emerges that both reduces computational cost and ensures high accuracy. This study also details the underlying mathematical principles of the optimal approximation provided by spline functions in Sobolev spaces; the connection between theory and application is supported by numerical experiments on sample PDE problems. The results obtained reveal that, compared to classical approaches, the same or better accuracy can be achieved with fewer degrees of freedom. In this way, it provides significant motivation for further development of spline-based methods in both theoretical and computational aspects for future research. As a result, this article aims to serve as an important guide for obtaining highly accurate and efficient solutions by offering new insights into the interaction of Sobolev spaces and spline functions in solving partial differential equations.