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arXiv 1408.0182math.NAcs.NA
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多区域不连续伽辽金IgA近似方法对椭圆边值问题的分析

Analysis of Multipatch Discontinuous Galerkin IgA Approximations to Elliptic Boundary Value Problems

Johann Radon Institute for Computational and Applied Mathematics (RICAM)（约翰·拉登计算与应用数学研究所）

Ulrich Langer, Ioannis Toulopoulos

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AI总结：

本文研究多区域不连续伽辽金IgA方法在求解具有间断扩散系数的线性扩散问题中的近似性质，通过B-样条或NURBS构造子域，并利用内罚项处理界面间断，证明了在二维和三维域中具有最优收敛率的离散误差分析。

摘要：

在本文中，我们研究了多区域不连续伽辽金IgA方法的近似性质，该方法应用了多区域IgA离散化概念和不连续伽辽金技术来处理不同区域之间的界面。该方法用于求解具有可能在区域界面处间断的扩散系数的线性扩散问题。计算域被划分为非重叠的子域，称为IgA中的区域，在这些子域中构造B-样条或NURBS有限维近似空间。在每个子域中近似问题的解，不施加任何匹配网格条件，也不对离散解在界面处的连续性提出要求。为了处理离散解在界面处的间断，应用了具有内罚项的数值流。我们为二维和三维域中的问题提供了严谨的先验离散误差分析，这些问题的解属于$W^{l,p}, l\geq 2,{\ } p\in ({2d}/{(d+2(l-1))},2]$。无论如何，我们展示了离散解相对于dG-范数的最优收敛率。

英文摘要：

In this work, we study the approximation properties of multi-patch dG-IgA methods, that apply the multipatch Isogeometric Analysis (IgA) discretization concept and the discontinuous Galerkin (dG) technique on the interfaces between the patches, for solving linear diffusion problems with diffusion coefficients that may be discontinuous across the patch interfaces. The computational domain is divided into non-overlapping sub-domains, called patches in IgA, where $B$-splines, or NURBS finite dimensional approximations spaces are constructed. The solution of the problem is approximated in every sub-domain without imposing any matching grid conditions and without any continuity requirements for the discrete solution across the interfaces. Numerical fluxes with interior penalty jump terms are applied in order to treat the discontinuities of the discrete solution on the interfaces. We provide a rigorous a priori discretization error analysis for problems set in 2d- and 3d- dimensional domains, with solutions belonging to $W^{l,p}, l\geq 2,{\ } p\in ({2d}/{(d+2(l-1))},2]$. In any case, we show optimal convergence rates of the discretization with respect to the dG - norm.

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