Mathematical analysis of variational isogeometric methods*

IF 16.3 1区数学 Q1 MATHEMATICS

Acta Numerica Pub Date : 2014-05-01 DOI:10.1017/S096249291400004X

L. Veiga, A. Buffa, G. Sangalli, Rafael Vázquez Hernández

引用次数: 250

Abstract

This review paper collects several results that form part of the theoretical foundation of isogeometric methods. We analyse variational techniques for the numerical resolution of PDEs based on splines or NURBS and we provide optimal approximation and error estimates in several cases of interest. The theory presented also includes estimates for T-splines, which are an extension of splines allowing for local refinement. In particular, we focus our attention on elliptic and saddle point problems, and we define spline edge and face elements. Our theoretical results are demonstrated by a rich set of numerical examples. Finally, we discuss implementation and efficiency together with preconditioning issues for the final linear system.

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变分等几何方法的数学分析*

这篇综述收集了几个结果，这些结果构成了等几何方法的一部分理论基础。我们分析了基于样条或NURBS的偏微分方程数值分辨率的变分技术，并在几个感兴趣的情况下提供了最佳逼近和误差估计。提出的理论还包括t样条的估计，它是样条的扩展，允许局部细化。重点研究了椭圆点和鞍点问题，定义了样条边和样条面元素。我们的理论结果通过一组丰富的数值算例得到了验证。最后，我们讨论了最终线性系统的实现和效率以及预处理问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Acta Numerica MATHEMATICS-

CiteScore

26.00

自引率

0.70%

发文量

期刊介绍： Acta Numerica stands as the preeminent mathematics journal, ranking highest in both Impact Factor and MCQ metrics. This annual journal features a collection of review articles that showcase survey papers authored by prominent researchers in numerical analysis, scientific computing, and computational mathematics. These papers deliver comprehensive overviews of recent advances, offering state-of-the-art techniques and analyses. Encompassing the entirety of numerical analysis, the articles are crafted in an accessible style, catering to researchers at all levels and serving as valuable teaching aids for advanced instruction. The broad subject areas covered include computational methods in linear algebra, optimization, ordinary and partial differential equations, approximation theory, stochastic analysis, nonlinear dynamical systems, as well as the application of computational techniques in science and engineering. Acta Numerica also delves into the mathematical theory underpinning numerical methods, making it a versatile and authoritative resource in the field of mathematics.