Adaptive High-Order Quarklet Frame Methods for Elliptic Operator Equations

椭圆算子方程的自适应高阶 Quarklet 框架方法

• 批准号: 451355735
• 负责人: Professor Dr. Stephan Dahlke
• 金额: --
• 依托单位: Department Mathematik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/451355735?language=en
• 关键词: Adaptive; Order; Quarklet; Frame; Methods

We are concerned with the design, convergence analysis and efficient realization of a new class of adaptive, high-order numerical methods for partial differential equations. We will consider basis-oriented schemes that work with a wavelet version of hp finite element dictionaries, so-called quarklet systems. These piecewise polynomial, oscillatory functions share the high-order approximation properties of hp FE systems, and they have the frame property in a variety of function spaces, including positive- and negative-order Sobolev spaces, thereby enabling anisotropic tensor product approximation techniques. In this project, we will exploit these approximation and stability properties of quarklet systems in order to derive adaptive discretization methods that converge at sub-exponential rates in many cases. We will explore a combination of new multiscale regularity estimation techniques, associated grid refinement schemes and adaptive space splittings. We intend to apply the resulting adaptive quarklet schemes to the numerical solution of elliptic differential equations and of parabolic evolution problems in a space-time, first-order systems least squares formulation. We expect that the convergence analysis of adaptive quarklet schemes can also help to foster a better understanding of hp finite element methods themselves.

我们关注的是一类新的自适应高阶偏微分方程数值方法的设计，收敛性分析和有效实现。我们将考虑面向基的方案，它与小波版本的hp有限元字典，即所谓的夸克系统一起工作。这些分段多项式，振荡函数共享的高阶逼近性能的HP FE系统，他们在各种功能空间，包括正，负阶Sobolev空间的框架属性，从而使各向异性张量积逼近技术。在这个项目中，我们将利用这些近似和稳定性的夸克系统，以获得自适应离散化方法，在许多情况下，收敛于次指数率。我们将探索新的多尺度规律性估计技术，相关的网格细化方案和自适应空间分裂的组合。我们打算将由此产生的自适应夸克计划的椭圆型微分方程的数值解和抛物演化问题的时空，一阶系统最小二乘制定。我们期望自适应夸克子格式的收敛性分析也有助于更好地理解hp有限元方法本身。