标题： 使用重叠张量积自由结点B样条的能量最小化 显示英文标题

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

摘要： 准确求解具有局部特征的偏微分方程需要细化网格，使其适应解。传统的数值方法，如有限元方法，本质上是线性的，对于此类问题通常效果不佳，因为网格并未针对解进行调整。自适应策略，如$h$-和$p$-细化，通过基于后验误差估计依次细化网格来提高效率。然而，这些方法在几何上是刚性的——仅限于特定的细化规则——并且需要在一系列自适应网格上求解问题，这可能计算成本较高。此外，设计有效的后验误差估计是依赖于问题的且不简单的。在本工作中，我们研究了一种基于重叠张量积自由结点B样条片的特定非线性逼近方案，其中结点位置作为控制离散化几何的非线性参数。我们分析了对应的能量最小化问题，针对线性、自伴随椭圆偏微分方程，表明在适度的网格尺寸条件下，离散能量满足约束优化方案所需的结构特性，该方案在我们的配套工作中开发[Magueresse, Badia (2025, arXiv:2508.17687)]。这建立了两种分析之间的直接联系：此处考虑的自适应自由结点B样条空间适用于抽象框架，确保了对结点位置和系数联合优化的投影梯度下降的收敛性。数值实验说明了该方法的效率及其能够以显著更少的自由度捕捉局部特征的能力。 显示英文摘要

摘要： 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.