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

Year 2026, Volume: 18 Issue: 1, 143 - 158, 23.02.2026

Aytekin Enver , Fatma Ayaz

https://doi.org/10.47000/tjmcs.1774647

https://izlik.org/JA48UN46WR

Abstract

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.

Keywords

Sobolev Spaces , Spline Approximation , Isogeometric Analysis (IGA) , Error Estimates; , Partial Differential Equations

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