Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps
Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps
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DOI：
10.1007/s00211-005-0588-3
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发表时间：
2005-04
影响因子：
2.1
通讯作者：
C. Bacuta;V. Nistor;L. Zikatanov
中科院分区：
文献类型：
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作者：
C. Bacuta;V. Nistor;L. Zikatanov
LetuanduV∈Vbe分别是非齐次Dirichlet问题的解和离散解Δu=fon a,u|∂a =0。对于任意∈n和任意有界多边形域，我们给出了一个有限维子空间序列的构造，其中∈m−1(a)是任意的，且ci是一个仅依赖于a而不依赖于n的常数（我们不假设∈m+1(a)）。这种子空间序列的存在性是由Babuška[8]在一篇开创性的论文中首先证明的。我们的方法不同；基于Sobolev空间的齐次性和适当Sobolev空间中非齐次Dirichlet问题的适定性，给出了一个新的证明，并代替了光滑边界域边值问题的常用“移位定理”。我们的结果立即推广到边界有圆锥点的区域。我们还指出了处理带有尖点的域所需的一些更改。我们的数值计算与理论结果一致。
LetuanduV∈Vbe the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem Δu=fon ℙ,u|∂ℙ=0. For anym∈ ℕ and any bounded polygonal domain ℙ, we provide a construction of a new sequence of finite dimensional subspacesVnsuch that wheref∈Hm−1(ℙ) is arbitrary andCis a constant that depends only on ℙ and not onn(we donotassumeu∈Hm+1(ℙ)). The existence of such a sequence of subspaces was first proved in a ground–breaking paper by Babuška [8]. Our method is different; it is based on the homogeneity properties of Sobolev spaces with weights and the well–posedness of non-homogeneous Dirichlet problem in suitable Sobolev spaces with weights, for which we provide a new proof, and which is a substitute of the usual “shift theorems” for boundary value problems in domains with smooth boundary. Our results extended right away to domains whose boundaries have conical points. We also indicate some of the changes necessary to deal with domains with cusps. Our numerical computation are in agreement with our theoretical results.