QUASI-OPTIMAL CONVERGENCE RATE OF AN ADAPTIVE DISCONTINUOUS GALERKIN METHOD
QUASI-OPTIMAL CONVERGENCE RATE OF AN ADAPTIVE DISCONTINUOUS GALERKIN METHOD
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DOI：
10.1137/08072838x
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发表时间：
2010-01-01
影响因子：
2.9
通讯作者：
Nochetto, Ricardo H.
中科院分区：
文献类型：
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作者：
Bonito, Andrea;Nochetto, Ricardo H.
研究了二阶对称线性椭圆算子的自适应间断有限元方法。该方法是制定的单形或四边形，与任何多项式的次数和在任何维度>= 2的三角形网格。我们证明了ADFEM是两个连续的自适应环路之间的能量误差和缩放误差估计的总和的收缩。我们设计了一个细化过程，保持一致有界的非一致性水平，并证明了使用连续和不连续有限元的近似类是等价的。几何衰减和类的等价性是推导ADFEM最优基数的工具。我们表明，ADFEM（和AFEM上的ESTA网格）产生的能量误差加上振荡的衰减率的自由度的数量，由最佳近似为这个组合的非线性量。
We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension >= 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the ADFEM. We show that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.