Energy minimisation using overlapping tensor-product free-knot B-splines

TL;DR

This paper introduces an energy minimisation approach using overlapping tensor-product free-knot B-splines for PDEs, enabling adaptive, efficient discretisation that captures localised features with fewer degrees of freedom.

Contribution

It develops a nonlinear approximation scheme with energy-based optimisation for free-knot B-splines, ensuring convergence and efficiency in solving PDEs with localised features.

Findings

Method captures localised features efficiently

Fewer degrees of freedom needed compared to finite elements

Convergence of the optimisation scheme is established

Abstract

Accurately solving PDEs with localised features requires refined meshes that adapt to the solution. Traditional numerical methods, such as finite elements, are linear in nature and often ineffective for such problems, as the mesh is not tailored to the solution. Adaptive strategies, such as $h$- and $p$-refinement, improve efficiency by sequentially refining the mesh based on a posteriori error estimates. However, these methods are geometrically rigid -- limited to specific refinement rules -- and require solving the problem on a sequence of adaptive meshes, which can be computationally expensive. Moreover, the design of effective a posteriori error estimates is problem-dependent and non-trivial. In this work, we study a specific nonlinear approximation scheme based on overlapping tensor-product free-knot B-spline patches, where knot positions act as nonlinear parameters controlling the geometry of the discretisation. We analyse the corresponding energy minimisation problem for linear, self-adjoint elliptic PDEs, showing that, under a mild mesh size condition, the discrete energy satisfies the structural properties required for the local and global convergence of the constrained optimisation scheme developed in our companion work [Magueresse, Badia (2025, arXiv:2508.17687)]. This establishes a direct connection between the two analyses: the adaptive free-knot B-spline space considered here fits into the abstract framework, ensuring convergence of projected gradient descent for the joint optimisation of knot positions and coefficients. Numerical experiments illustrate the method's efficiency and its ability to capture localised features with significantly fewer degrees of freedom than standard finite element discretisations.