A Local Error Estimate for the Poisson Equation with a Line Source Term
A Local Error Estimate for the Poisson Equation with a Line Source Term
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具有线源项的泊松方程的局部误差估计
DOI：
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发表时间：
2015
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通讯作者：
B. Wohlmuth
中科院分区：
文献类型：
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作者：
T. Köppl;Ettore Vidotto;B. Wohlmuth
在本文中，我们展示了三个空间维度（3D）中泊松方程的局部先验误差估计，其中源项是集中在一条线上的狄拉克测度。此类问题在许多应用领域中都可以找到。例如，在医学工程中，毛细血管和组织中的血流可以通过使用线源项耦合泊肃叶定律和达西定律来建模。由于线源项引起的奇异性，有限元解在经典范数下收敛得不是最优的。然而，奇点处的误差通常要么由模型误差主导（例如，在降维设置中），要么不是感兴趣的量（例如，在最优控制问题中）。因此，我们对局部误差估计感兴趣，即，我们在空间中考虑固定子域上的 L2 范数，不包括狄拉克测度集中的直线邻域。结果表明，线性有限元在这种范数下最优收敛到对数因子。一些数值测试证实了理论考虑。
In this paper, we show a local a priori error estimate for the Poisson equation in three space dimensions (3D), where the source term is a Dirac measure concentrated on a line. This type of problem can be found in many application areas. In medical engineering, e.g., blood flow in capillaries and tissue can be modeled by coupling Poiseuille’s and Darcy’s law using a line source term. Due to the singularity induced by the line source term, finite element solutions converge suboptimal in classical norms. However, quite often the error at the singularity is either dominated by model errors (e.g. in dimension reduced settings) or is not the quantity of interest (e.g. in optimal control problems). Therefore we are interested in local error estimates, i.e., we consider in space a L2-norm on a fixed subdomain excluding a neighborhood of the line, where the Dirac measure is concentrated. It is shown that linear finite elements converge optimal up to a log-factor in such a norm. The theoretical considerations are confirmed by some numerical tests.