This paper is concerned with new discretization methods for the numerical treatment of elliptic partial differential equations. We derive an adaptive approximation scheme that is based on frames of quarkonial type, which can be interpreted as a wavelet version of |$hp$| finite element dictionaries. These new frames are constructed from a finite set of functions via translation, dilation and multiplication by monomials. By using nonoverlapping domain decomposition ideas, we establish quarkonial frames on domains that can be decomposed into the union of parametric images of unit cubes. We also show that these new representation systems are stable in a certain range of Sobolev spaces. The construction is performed in such a way that, similar to the wavelet setting, the frame elements, the so-called quarklets, possess a certain number of vanishing moments. This enables us to generalize the basic building blocks of adaptive wavelet algorithms to the quarklet case. The applicability of the new approach is demonstrated by numerical experiments for the Poisson equation on |$L$|-shaped domains.

中文翻译：

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椭圆型偏微分方程的自适应四方域分解方法

本文涉及椭圆偏微分方程数值处理的新离散化方法。我们推导了一种基于近似类型帧的自适应近似方案，可以将其解释为| $ hp $ |的小波形式。有限元字典。这些新框架是通过有限的一组函数通过单项式的平移，扩张和乘法来构造的。通过使用非重叠域分解思想，我们在可以分解为单位立方体的参数图像并集的域上建立了等距框架。我们还表明，这些新的表示系统在一定范围的Sobolev空间中是稳定的。构造的执行方式类似于小波设置，即框架元素，即所谓的框架元素。夸克，具有一定数量的消失时刻。这使我们能够将自适应小波算法的基本构建模块推广到四重奏情况。通过对| $ L $ |的Poisson方程进行数值实验，证明了该新方法的适用性。形域。