Adaptive High-Order Quarklet Frame Methods for Elliptic Operator Equations

我们关注一类新的自适应高阶偏微分方程数值方法的设计、收敛分析和有效实现。我们将考虑与小波版本的hp有限元字典一起工作的面向基的方案，即所谓的夸克系统。这些分段多项式振荡函数共享hp有限元系统的高阶近似性质，并且它们在各种函数空间（包括正阶和负阶Sobolev空间）中具有框架性质，从而实现各向异性张量积近似技术。在这个项目中，我们将利用夸克系统的这些近似和稳定性特性来推导在许多情况下以次指数速率收敛的自适应离散化方法。我们将探索新的多尺度规则估计技术，相关网格细化方案和自适应空间分割的组合。我们打算将由此产生的自适应夸克格式应用于椭圆微分方程和抛物演化问题的时空一阶系统最小二乘公式的数值解。我们期望自适应夸克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。

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.