各向异性局部重构型后验误差估计及自适应计算

发布时间：2018-04-17 20:10

本文选题：后验误差估计 + 各向异性网格　；参考：《郑州大学》2016年博士论文

【摘要】：本文主要针对带间断系数的扩散问题和奇异摄动反应-扩散问题,在各向异性网格条件下推导健壮的局部重构型后验误差估计,进而实现各向异性自适应计算。针对带间断系数的扩散问题,我们推导了适用于各向异性网格的局部重构型后验误差估计。基于平衡通量的局部守恒假设,一个抽象的误差上界首先被建立。剩下的任务就是构造满足假设的平衡通量。这里,我们给出了平衡通量的两种构造方式：直接法和混合有限元法。其中直接法是最容易想到的,对网格各向异性的影响不需要特别的处理,而混合有限元法需要将网格的各向异性尺寸引入到平衡通量的构造中。这类后验估计关于扩散系数是健壮的,且可应用于有限体积法,有限差分法和有限元法。我们分别在能量范数和对偶范数意义下表明了所构造的误差估计子是可靠的、有效的。事实上,这些估计在各向同性网格条件下与Vohralik (J Sci Comput 46:397-438,2011)得到的各向同性后验估计等价,因此从某种意义上来说,我们推广了Vohralik的结果。针对奇异摄动反应-扩散问题的有限体积法,我们也推导了适用于各向异性网格的局部重构型后验误差估计。类似地,基于平衡通量的局部守恒假设,我们首先在能量范数意义下建立了一个抽象的误差上界。这个平衡通量的构造采用直接法。我们在能量范数意义下表明了所构造的误差估计子是可靠的、有效的且关于反应系数是健壮的。事实上,这些估计在各向同性网格条件下与Cheddadi等人(ESAIM-Math Model Numer Anal 43:867-888,2009)得到的各向同性后验估计等价,因此从某种意义上来说,我们推广了Cheddadi等人的结果。基于所得的各向异性后验误差估计,我们推导出相应的各向异性误差指示子,从而用于引导网格的各向异性自适应加密。事实上,原始形式的各向异性后验估计并不能直接用于引导网格的各向异性自适应加密,因为它不能表示出误差在各个方向的贡献,且对齐测度中含有未知的真解。为了得到各向异性误差指示子,我们改写了对齐测度的格式,并通过后处理的方式得到它的一个逼近。最后,我们给出了一个各向异性网格自适应加密算法,能够利用所得到的各向异性误差指示子生成一个合适的度量场,从而生成一个新的各向异性网格。以上理论结果都已得到数值实验的验证。
[Abstract]:In this paper, for the diffusion problem with discontinuous coefficients and the singularly perturbed reaction-diffusion problem, robust locally reconstructed posteriori error estimates are derived under anisotropic mesh conditions, and then anisotropic adaptive computation is realized.For the diffusion problem with discontinuous coefficients, we derive a locally reconstructed posteriori error estimate for anisotropic meshes.Based on the local conservation hypothesis of equilibrium flux, an abstract upper bound of error is first established.The remaining task is to construct an equilibrium flux that meets the assumptions.Here, we give two ways of constructing equilibrium flux: direct method and mixed finite element method.The direct method is the most easy to consider, and the influence on the anisotropy of the mesh does not need special treatment, while the hybrid finite element method needs to introduce the anisotropic size of the mesh into the construction of the equilibrium flux.This kind of posterior estimation is robust on diffusion coefficient and can be applied to finite volume method, finite difference method and finite element method.We show that the constructed error estimators are reliable and effective in the sense of energy norm and dual norm respectively.In fact, these estimates are equivalent to the isotropic posteriori estimates obtained by Vohralik J Sci Comput 46: 397-438 (2011) under the isotropic grid condition. Therefore, in some sense, we generalize the results of Vohralik.For singularly perturbed reaction-diffusion problems, we also derive a locally reconstructed posteriori error estimate for anisotropic meshes.Similarly, based on the local conservation hypothesis of equilibrium flux, we first establish an abstract error upper bound in the sense of energy norm.This equilibrium flux is constructed by direct method.In the sense of energy norm, we show that the constructed error estimator is reliable, effective and robust with respect to the response coefficient.In fact, these estimates are equivalent to the isotropic posteriori estimates obtained by Cheddadi et al. (ESAIM-Math Model Numer Anal 43: 867-888 / 2009), so in a sense we generalize the results of Cheddadi et al.Based on the obtained anisotropic posteriori error estimation, we derive the corresponding anisotropic error indicator, which can be used to guide the anisotropic adaptive encryption of the mesh.In fact the original form of anisotropic posteriori estimation can not be directly used to guide the anisotropic adaptive encryption of the mesh because it can not represent the contribution of errors in all directions and the alignment measure contains the unknown true solution.In order to obtain the anisotropic error indicator, we reformulate the format of the alignment measure and obtain an approximation by post-processing.Finally, we present an adaptive encryption algorithm for anisotropic meshes, which can generate an appropriate metric field using the anisotropic error indicator, thus generating a new anisotropic mesh.The above theoretical results have been verified by numerical experiments.
【学位授予单位】：郑州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.82

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