Energy Norm A-Posteriori Error Estimates for a Discontinuous Galerkin Scheme Applied to Elliptic Problems with an Interface
Energy Norm A-Posteriori Error Estimates for a Discontinuous Galerkin Scheme Applied to Elliptic Problems with an Interface
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应用于带界面椭圆问题的不连续伽辽金方案的能量范数后验误差估计
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发表时间：
2009
期刊：
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通讯作者：
P. Zunino
中科院分区：
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作者：
P. Zunino
众所周知，具有界面的二阶椭圆问题的解可能具有内层和/或奇点。我们提出了一个自适应间断伽辽金（DG）方法，适当地近似这样的问题。首先，我们介绍了加权内部惩罚方法，它推广了经典的内部惩罚DG计划取代算术平均与适当的加权平均权取决于系数的问题。然后，我们讨论了一个家庭的建设的残差为基础的局部误差指标的能量范数，适用于对流扩散反应方程的扩散参数，可能是不连续的沿着接口。特别是，我们演示了如何将权重纳入计划的一些先验知识的精确解，提高了效率的估计和相应的适应网格。数值实验验证了理论结果。
It is well known that the solution of second order elliptic problems with interfaces may feature internal layers and/or singularities. We present an adaptive discontinuous Galerkin (DG) method to suitably approximate such problems. First, we introduce the weighted interior penalty method, which generalizes the classical interior penalty DG schemes by replacing the arithmetic means with suitably weighted averages where the weights depend on the coefficients of the problem. Then, we discuss the construction of a family of residual based local error indicators for the energy norm, applied to advection-diffusion-reaction equations featuring a diffusivity parameter that may be discontinuous along an interface. In particular, we demonstrate how the weights can incorporate into the scheme some a-priori knowledge of the exact solution that improves the efficacy of the estimator and of the corresponding adapted mesh. The theoretical results are confirmed by means of numerical experiments.