An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis.
An unfitted hybridizable discontinuous Galerkin method for the Poisson interface problem and its error analysis.
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泊松界面问题的不拟合杂化间断伽辽金方法及其误差分析
DOI：
10.1093/imanum/drv071
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Z. Q. Xie
中科院分区：
文献类型：
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作者：
H. X. Dong;B. Wang;Z. Q. Xie
在本文中，我们提出并分析了泊松接口问题的不拟合网格方法。通过在界面附近构造一个新的 ansatz 函数，我们能够导出一个扩展的泊松问题，其界面适合给定的准均匀三角网格。然后我们采用可混合的不连续伽辽金方法来解决扩展问题，并选择适当的通量来处理跳跃条件。与现有方法相比，ansatz 函数是通过精细的分段二次 Hermite 多项式插值设计的，并通过标准拉格朗日多项式插值进行后处理。这种显式函数为任何形状的界面的基础解的奇异部分提供了三阶近似。这对于所提出方法的稳定性和收敛性也至关重要。此外，我们提供了严格的误差分析，表明该方案对于解及其梯度的逼近可以达到二阶收敛速度。具有复杂接口的大量数值示例证明了该方法的预期收敛顺序和鲁棒性。
In this article, we present and analyse an unfitted mesh method for the Poisson interface problem. By constructing a novel ansatz function in the vicinity of the interface, we are able to derive an extended Poisson problem whose interface fits a given quasi-uniform triangular mesh. Then we adopt a hybridizable discontinuous Galerkin method to solve the extended problem with an appropriate choice of flux for treating the jump conditions. In contrast with existing approaches, the ansatz function is designed through a delicate piecewise quadratic Hermite polynomial interpolation with a post-processing via a standard Lagrange polynomial interpolation. Such an explicit function offers a third-order approximation to the singular part of the underlying solution for interfaces of any shape. It is also essential for both stability and convergence of the proposed method. Moreover, we provide rigorous error analysis to show that the scheme can achieve a second-order convergence rate for the approximation of the solution and its gradient. Ample numerical examples with complex interfaces demonstrate the expected convergence order and robustness of the method.