Error reduction and convergence for an adaptive mixed finite element method
Error reduction and convergence for an adaptive mixed finite element method
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DOI：
10.1090/s0025-5718-06-01829-1
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发表时间：
2006-03
期刊：
影响因子：
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通讯作者：
C. Carstensen;R. Hoppe
中科院分区：
文献类型：
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作者：
C. Carstensen;R. Hoppe
自适应混合有限元方法 (AMFEM) 旨在保证误差减少，也称为饱和特性：在每个细化步骤之后，细网格的误差严格小于粗网格的误差，直至振荡项。此处针对 Raviart-Thomas 有限元方法建立了此误差减少特性，对于通量误差的 L 2 范数，统一减少因子 p < 1。我们的结果允许适当的自适应混合有限元算法相对于细化级别的数量进行线性收敛。令人惊讶的是，与一致的有限元方法不同，自适应算法不需要任何特定的网格设计。新的论点是离散局部效率和准正交估计。该证明不依赖于对偶性或正则性。
An adaptive mixed finite element method (AMFEM) is designed to guarantee an error reduction, also known as saturation property: after each refinement step, the error for the fine mesh is strictly smaller than the error for the coarse mesh up to oscillation terms. This error reduction property is established here for the Raviart-Thomas finite element method with a reduction factor p < 1 uniformly for the L 2 norm of the flux errors. Our result allows for linear convergence of a proper adaptive mixed finite element algorithm with respect to the number of refinement levels. The adaptive algorithm surprisingly does not require any particular mesh design, unlike the conforming finite element method. The new arguments are a discrete local efficiency and a quasi-orthogonality estimate. The proof does not rely on duality or on regularity.