HIGH-ORDER GALERKIN APPROXIMATIONS FOR PARAMETRIC SECOND-ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
HIGH-ORDER GALERKIN APPROXIMATIONS FOR PARAMETRIC SECOND-ORDER ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
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参数二阶椭圆偏微分方程的高阶伽略金近似
DOI：
10.1142/s0218202513500218
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发表时间：
2013
影响因子：
3.5
通讯作者：
C. Schwab
中科院分区：
文献类型：
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作者：
V. Nistor;V. Nistor;C. Schwab
设D_∞（？）d，d = 2，3，是具有分片光滑边界的有界区域，Y = 1 ∞（？），U = B_1（Y）是Y的开单位球.考虑D上的参数一致强椭圆二阶偏微分算子族（Py）y∈U.在适当的系数假设下，我们建立了参数边值问题Py u（x，y）= f（x，y），x ∈ D，y ∈ U的解u的正则性结果，其中混合Dirichlet-Neumann边值条件分别在D上和D上.我们的正则性和适定性结果是在Kondrat'ev型加权Sobolev空间的尺度上得到的。我们证明了（Py）y ∈ U存在一个移位定理，该定理在参数y ∈ U上是一致的.具体地说，如果P的系数满足y =（yk）k≥1 ∈ U，并且如果序列在k中p-可和，对于0 < p< 1，则参数解u允许展开成张量化勒让德多项式Lν（y），使得相应的序列，其中。我们还显示了最佳的代数收敛阶的Galerkin近似UL的解决方案u使用合适的有限元空间在二维和三维。即，设t = m/d，s = 1/p-1/2，其中0 < p < 1。证明了对任意m ∈ N，存在一个嵌套的有限维空间Sl <$L2（U;V）序列{Sl}l≥0，使得u的Galerkin投影ul ∈ Sl满足<$u-ul <$L2（U;V）≤ C dim（Sl）-min{s，t}<$f <$Hm-1（D），dim（Sl）→ ∞.序列Sl是使用D中的有限元空间序列Vμ <$V构造的，具有朝向奇点的分级网格细化。每个子空间Sl由u的Legendre混沌展开中的“活动多项式混沌”系数uν ∈ V，ν ∈ Λl的有限子集定义，对于每个ν ∈ Λl，适当地选择μ（l，ν），所述系数u ν ∈ Vμ（l，ν）近似为vν ∈ Vμ（l，ν）。
Let D ⊂ ℝd, d = 2, 3, be a bounded domain with piecewise smooth boundary, Y = l∞(ℕ) and U = B1(Y), the open unit ball of Y. We consider a parametric family (Py)y∈U of uniformly strongly elliptic, second-order partial differential operators Py on D. Under suitable assumptions on the coefficients, we establish a regularity result for the solution u of the parametric boundary value problem Py u(x, y) = f(x, y), x ∈ D, y ∈ U, with mixed Dirichlet–Neumann boundary conditions on ∂d D and, respectively, on ∂n D. Our regularity and well-posedness results are formulated in a scale of weighted Sobolev spaces of Kondrat'ev type. We prove that the (Py)y ∈ U admit a shift theorem that is uniform in the parameter y ∈ U. Specifically, if the coefficients of P satisfy , y = (yk)k≥1 ∈ U and if the sequences are p-summable in k, for 0 < p< 1, then the parametric solution u admits an expansion into tensorized Legendre polynomials Lν(y) such that the corresponding sequence , where . We also show optimal algebraic orders of convergence for the Galerkin approximations ul of the solution u using suitable Finite Element spaces in two and three dimensions. Namely, let t = m/d and s = 1/p-1/2, where , 0 < p < 1. We show that, for each m ∈ ℕ, there exists a sequence {Sl}l≥0 of nested, finite-dimensional spaces Sl ⊂ L2(U;V) such that the Galerkin projections ul ∈ Sl of u satisfy ‖u - ul‖L2(U;V) ≤ C dim(Sl)-min{s, t} ‖f‖Hm-1(D), dim(Sl) → ∞. The sequence Sl is constructed using a sequence Vμ⊂V of Finite Element spaces in D with graded mesh refinements toward the singularities. Each subspace Sl is defined by a finite subset of "active polynomial chaos" coefficients uν ∈ V, ν ∈ Λl in the Legendre chaos expansion of u which are approximated by vν ∈ Vμ(l, ν), for each ν ∈ Λl, with a suitable choice of μ(l, ν).
《极端条件下的XMCD研究——巡回电子系统中的磁相变——》（特邀）
DOI：
--
发表时间：
2007
期刊：
影响因子：
--
作者：
H.;Maruyama
通讯作者：
Maruyama