Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D
Anisotropic graded meshes and quasi-optimal rates of convergence for the FEM on polyhedral domains in 3D
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发表时间：
2012
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通讯作者：
C. Bacuta;Hengguang Li;V. Nistor
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作者：
C. Bacuta;Hengguang Li;V. Nistor
我们考虑模型问题−∆u + V u = f∈Ω，在∂Ω上具有合适的边界条件，其中Ω是R， d = 2,3中的有界多面体域，V是一个可能的奇异势。我们研究了该问题的有效有限元离散化。功能。分析的。Optim。机械工程学报，28(7-8):775-824,2007，牛。数学。Soc。科学。数学。罗曼尼（N.S.）， 55(103): 157-178, 2012，以及其他一些更近期的论文。在一些附加的温和假设下，我们展示了如何构造网格序列，使得Galerkin近似序列uk∈Sk在‖u−uk‖H1（Ω）≤C dim（Sk）‖f‖Hm−1（Ω）的意义上达到h-拟最优收敛率。定义有限元空间Sk的网格适当地向奇异点渐变。它们与均匀细化得到的网格在拓扑上是等价的，因此它们的构造容易实现。我们详细解释了四个典型问题的网格细化：V = 0的混合边值/传输问题，二维多边形域上具有平方反势V的Schrödinger型算子，三维域上具有周期平方反势V的Schrödinger型算子，以及三维域上的泊松问题。这些问题按复杂程度的递增顺序列出。这里考虑的传输问题包括多个接点的情况。Constantin Băcuţă，李恒光，Victor Nistor引言设Ω∧R是一个开的有界集合。考虑在有界域Ω∧R上定义的Ω(1)中的模型问题（−∆+ V）u = f，其中∆是拉普拉斯函数∆=∑d i=1∂2 i。将施加适当的边界条件。这个模型问题（即使对于V = 0）在许多实际应用中都会出现。我们感兴趣的是寻找解u近似的有效数值方法。在本文中，我们将处理经典的有限元方法。u的有限元近似uk的收敛速度取决于（至少对于准均匀网格）解u的光滑性。当∂Ω是光滑的并且V = 0时，众所周知，我们的模型问题(1)对于任何f∈Hm−1（Ω）和g∈H（∂Ω）[6]具有唯一解u∈H（Ω），其中g是Dirichlet边界条件，u = g在∂Ω上。此外,u不断取决于f和g。这个结果是经典的平稳泊松问题的适定性问题域和提供了一个令人满意的平滑结果u。另一方面,当Ω不光滑和V = 0,这也是众所周知的(4、5、9、10、11),存在s = sΩ,u∈H(Ω)对于任何年代0,独立于k和f,这样为u−英国为H1(Ω)≤C暗(Sk)为f为Hm−1(Ω),英国∈Sk,(2)其中d = 2或d = 3是我们多面体域Ω的维度。找到uk所需的工作量取决于Sk的维度，因此很自然地将误差与dim（Sk）进行比较，特别是因为通常的参数h对于非准均匀网格序列失去了意义。我们的主要目标是定义合适的子空间Sk∧H（Ω），满足我们问题中的所有基本边界条件，使得近似f∈Hm−1（Ω）的方程(1)的解u的伽辽金有限元投影uk∈Sk达到H -拟最优收敛速率。我们将在摘要中提到的四种情况下展示如何做到这一点：1)二维中V = 0的混合边值/传输问题，2)二维多边形域上具有平方反势V的Schrödinger型算子，3)三维中具有周期性平方反势V的Schrödinger型算子，以及4)三维域上的泊松问题（因此V = 0）。以上案例按复杂程度的递增顺序排列。这里考虑的二维传输问题包括多个连接点的情况。
We consider the model problem −∆u + V u = f ∈ Ω, with suitable boundary conditions on ∂Ω, where Ω is a bounded polyhedral domain in R, d = 2, 3, and V is a possibly singular potential. We study efficient finite element discretizations of our problem following Numer. Funct. Anal. Optim., 28(7-8):775–824, 2007, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 55(103):157–178, 2012, and a few other more recent papers. Under some additonal mild assumptions, we show how to construct sequences of meshes such that the sequence of Galerkin approximations uk ∈ Sk achieve h-quasi-optimal rates of convergence in the sense that ‖u − uk‖H1(Ω) ≤ C dim(Sk)‖f‖Hm−1(Ω). Our meshes defining the Finite Element spaces Sk are suitably graded towards the singularities. They are topologically equivalent to the meshes obtained by uniform refinement and hence their construction is easy to implement. We explain in detail the mesh refinement for four typical problems: for mixed boundary value/transmission problems for which V = 0, for Schrödinger type operators with an inverse square potential V on a polygonal domain in 2D, for Schrödinger type operators with a periodic inverse square potential V in 3D, and for the Poisson problem on a three dimensional domain. These problems are listed in the increasing order of complexity. The transmission problems considered here include the cases of multiple junction points. Constantin Băcuţă, Hengguang Li, and Victor Nistor Introduction Let Ω ⊂ R be an open, bounded set. Consider the model problem (−∆ + V )u = f in Ω (1) defined on a bounded domain Ω ⊂ R, where ∆ is the Laplacian ∆ = ∑d i=1 ∂ 2 i . Suitable boundary conditions will be imposed. This model problem (even for V = 0) arises in many practical applications. We are interested in finding efficient numerical methods for the approximation of the solution u. In this paper we shall deal with the classical Finite Element Method. The rate of convergence of the Finite Element approximations uk of u depends (at least for quasi-uniform meshes) on the smoothness of the solution u. When ∂Ω is smooth and V = 0, it is well known that our model problem (1) has a unique solution u ∈ H(Ω) for any f ∈ Hm−1(Ω) and g ∈ H(∂Ω) [6], where g is the Dirichlet boundary condition, u = g on ∂Ω. Moreover, u depends continuously on f and g. This result is the classical well-posedness of the Poisson problem on smooth domains and provides a satisfactory smoothness result for u. On the other hand, when Ω is not smooth and V = 0, it is also well known [4, 5, 9, 10, 11] that there exists s = sΩ such that u ∈ H(Ω) for any s 0, independent of k and f , such that ‖u− uk‖H1(Ω) ≤ C dim(Sk)‖f‖Hm−1(Ω), uk ∈ Sk, (2) where d = 2 or d = 3 is the dimension of our polyhedral domain Ω. The amount of work required to find uk depends on the dimension of Sk, so it is natural to compare the error with dim(Sk), especially since the usual parameter h loses its meaning for a non quasi-uniform sequence of meshes. Our main goal is to define suitable subspaces Sk ⊂ H(Ω), satisfying all essential boundary conditions in our problem, such that the Galerkin finite element projections uk ∈ Sk that approximate the solution u of Equation (1) for f ∈ Hm−1(Ω) achieve h-quasioptimal rates of convergence. We shall show how this is done in the four cases mentioned in the abstract: 1) for mixed boundary value/transmission problems with V = 0 in 2D, 2) for Schrödinger type operators with inverse square potential V on a polygonal domain in 2D, 3) for Schrödinger type operators with a periodic inverse square potential V in 3D, and 4) for the Poisson problem (so V = 0) on a three dimensional domain. The above cases were listed in the increasing order of complexity. The transmission problems considered here in2D include the cases of multiple junction points.