Adaptive Frame Methods for Elliptic Operator Equations: The Steepest Descent Approach

Adaptive Frame Methods for Elliptic Operator Equations: The Steepest Descent Approach
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椭圆算子方程的自适应框架方法：最速下降法

DOI：
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发表时间：
2007
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通讯作者：
R. Stevenson
R. Stevenson
中科院分区：
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文献类型：
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作者：
S. Dahlke;T. Raasch;M. Werner;M. Fornasier;R. Stevenson

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摘要：本文关注椭圆算子方程自适应数值方法的发展。我们对基于小波框架的离散化方案特别感兴趣。我们表明，通过使用三个基本子程序，可以导出可实现的收敛方案，而且该方案具有最佳的计算复杂度。该方案基于自适应最速下降迭代。我们通过计算一维和二维 L 形域区间上具有有限 Sobolev 平滑度的泊松方程解的数值结果来说明我们的发现。
Abstract: This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent iterations. We illustrate our findings by numerical results for the computation of solutions of the Poisson equation with limited Sobolev smoothness on intervals in 1D and L-shaped domains in 2D.