Adaptive High-Order Quarklet Frame Methods for Elliptic Operator Equations

点击翻译按钮获取中文摘要

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.