Anisotropic "Goal-Oriented" Mesh Adaptivity for Elliptic Problems
Anisotropic "Goal-Oriented" Mesh Adaptivity for Elliptic Problems
复制标题
椭圆问题的各向异性“面向目标”网格自适应性
DOI：
10.1137/120874606
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
R. Bermejo
中科院分区：
文献类型：
--
作者：
J. Carpio;J. Prieto;R. Bermejo
本文提出了一种各向异性自适应有限元算法，用于求解具有强各向异性特征的稳定线性平流扩散反应问题。误差分析基于对偶加权残差方法，允许我们对解决方案的某个函数$J(u)$进行“目标导向”的适应，并推导出具有线性和二次有限元的局部网格适应的“最佳”度量张量。作为一种新颖的方法，为了在由各向异性三角形组成的非结构化网格上评估误差估计器的权重，我们使用了一种易于扩展到任意阶有限元的补丁式高阶插值恢复。为了证明目标导向自适应方法的有效性，我们在二维空间中进行了大量的数值实验。我们计算了该解的一系列输出函数的收敛率和有效性指标。实验结果表明，该算法在线性和二次元情况下都具有良好的性能。
We propose in this paper an anisotropic, adaptive, finite element algorithm for steady, linear advection-diffusion-reaction problems with strong anisotropic features. The error analysis is based on the dual weighted residual methodology, allowing us to perform “goal-oriented” adaptation of a certain functional $J(u)$ of the solution and derive an “optimal” metric tensor for local mesh adaptation with linear and quadratic finite elements. As a novelty, and to evaluate the weights of the error estimator on unstructured meshes composed of anisotropic triangles, we make use of a patchwise, higher-order interpolation recovery readily extendable to finite elements of arbitrary order. We carry out a number of numerical experiments in two dimensions so as to prove the capabilities of the goal-oriented adaptive method. We compute the convergence rate and the effectivity index for a series of output functionals of the solution. The results show the good performance of the algorithm with linear as well as quadratic ...