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arXiv 1402.2117math.NAcs.NA
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适应性不连续伽辽金方法在曲面上的应用

Adaptive discontinuous Galerkin methods on surfaces

University of Warwick(华威大学);Mathematics Institute and Centre for Scientific Computing(数学研究所与科学计算中心)

Andreas Dedner, Pravin Madhavan

AI总结:

本文提出了一种基于残差的后验误差估计,用于在三维空间中隐式表示的光滑曲面上线性二阶椭圆问题的不连续伽辽金近似,分析了能量范数误差的残差部分和几何部分,并展示了该估计器在复杂曲面上的高效性。

摘要:

我们提出了一种双加权残差基于的后验误差估计,用于在三维空间中隐式表示的光滑曲面上线性二阶椭圆问题的不连续伽辽金(DG)近似。我们证明误差在能量范数中可以分为一个“残差部分”和更高阶的“几何部分”。证明了所得到的后验误差估计器的上界和下界,并考虑了多个具有挑战性的测试问题以展示估计器的可靠性和效率。我们还提出了一种新的“几何”驱动细化策略用于曲面上的PDEs,该策略显著提高了方法在复杂曲面上的性能。

英文摘要:

We present a dual weighted residual-based a posteriori error estimate for a discontinuous Galerkin (DG) approximation of a linear second-order elliptic problem on compact smooth connected and oriented surfaces in $\mathbb{R}^{3}$ which are implicitly represented as level sets of a smooth function. We show that the error in the energy norm may be split into a "residual part" and a higher order "geometric part". Upper and lower bounds for the resulting a posteriori error estimator are proven and we consider a number of challenging test problems to demonstrate the reliability and efficiency of the estimator. We also present a novel "geometric" driven refinement strategy for PDEs on surfaces which considerably improves the performance of the method on complex surfaces.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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