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首页> 外文期刊>Computer Methods in Applied Mechanics and Engineering >A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion
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A parallel and adaptive hybridized discontinuous Galerkin method for anisotropic nonhomogeneous diffusion

机译：各向异性非均匀扩散的并行自适应杂交不连续Galerkin方法

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摘要

The diffusion equation in anisotropic and nonhomogeneous media arises in the study of flow through porous media with sharp material interfaces. We discuss the solution of this problem by a hybrid discontinuous Galerkin (HDG) method. The method can be applied in three steps. First, we use a condensation technique to derive the scalar variable and the flux inside each element in terms of the numerical trace on the faces of that element. Then we form a global system of equations to solve for these numerical traces. We then solve the equation inside each element for the internal unknowns using the obtained numerical traces in the global solve step. Similar to other DG variants, HDG is a locally conservative method, and a noticeable share of calculations are performed independently within each element. In a mesh with pth order elements (p >= 0), this method gives p + 1 order of accuracy for smooth solutions for both the scalar variable and the flux. Moreover, by using a simple post-processing technique, one can reach an accuracy of order p + 2 for the scalar variable. To be able to handle problems with sharp material discontinuities, we use an adaptive refinement strategy, with octree grid structure. Hence, we avoid a larger global system which would arise from a fine uniform grid. In this process, we refine those elements with highest gradient of flux, at two sides of their faces. We have also utilized shared and distributed memory parallelism to enhance the performance of the method. The method is implemented using different modules of deal. II (Bangerth and Kanschat, 1999), PETSc (Balay et al., 2015), p4est (Burstedde et al., 2011) and Hypre (Falgout and Yang, 2002). To demonstrate the accuracy, efficiency, scalability, and flexibility of the method, several two and three dimensional numerical experiments are studied. (C) 2016 Elsevier B.V. All rights reserved.
机译：各向异性和非均匀介质中的扩散方程是在研究具有尖锐材料界面的多孔介质中流动时产生的。我们讨论通过混合不连续伽勒金（HDG）方法解决此问题。该方法可以分三个步骤应用。首先，我们使用缩合技术根据每个元素的表面上的数字轨迹导出每个元素内部的标量变量和通量。然后，我们形成一个全局方程组来求解这些数字迹。然后，我们使用全局求解步骤中获得的数值轨迹来求解内部未知数的每个元素内部的方程。与其他DG变体类似，HDG是一种局部保守方法，并且在每个元素内独立执行显着的计算份额。在具有p阶元素（p> = 0）的网格中，此方法为标量变量和通量的平滑解提供p + 1阶精度。此外，通过使用一种简单的后处理技术，对于标量变量，可以达到p + 2阶的精度。为了能够处理尖锐的材料间断问题，我们使用八叉树网格结构的自适应细化策略。因此，我们避免了由精细均匀的网格产生的更大的全局系统。在此过程中，我们优化了在其面的两侧具有最大磁通梯度的那些元素。我们还利用共享和分布式内存并行性来增强该方法的性能。该方法是使用不同的交易模块实现的。 II（Bangerth和Kanschat，1999），PETSc（Balay等，2015），p4est（Burstedde等，2011）和Hypre（Falgout和Yang，2002）。为了证明该方法的准确性，效率，可扩展性和灵活性，研究了几个二维和三维数值实验。 （C）2016 Elsevier B.V.保留所有权利。

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