The DUNE-DPG library for solving PDEs with Discontinuous Petrov--Galerkin finite elements
The DUNE-DPG library for solving PDEs with Discontinuous Petrov--Galerkin finite elements
复制标题
用于求解不连续 Petrov-Galerkin 有限元偏微分方程的 DUNE-DPG 库
DOI：
10.11588/ans.2017.1.27719
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发表时间：
2016
期刊：
影响因子：
--
通讯作者：
Olga Mula
中科院分区：
文献类型：
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作者：
Felix Gruber;Angela Klewinghaus;Olga Mula
在偏微分方程组的数值求解中，一个核心问题是构造不仅在无限维上而且在有限维上都是超稳定的变分公式。这保证了残差可以用来从下到上紧密地约束误差，并且对于后验误差控制和自适应策略的发展是至关重要的。在这个框架中，所谓的不连续Petrov-Galerkin(DPG)概念可以被视为设计具有这些期望的稳定性性质的变分公式的系统策略，见例如Broersen等人。[2015][2015]。在本文中，我们介绍了一个C++库Dune-DPG，它用于实现和求解这种变分公式。该库建立在多用途有限元程序包Dune上(见Blatt等人。[2016]))。Dune-DPG的主要特点之一是其灵活性，这是通过高度模块化的结构实现的。该库可以在实际中求解一些重要的偏微分方程组(其范围超出了经典的二阶椭圆型问题，包括例如运输占优问题)。因此，Dune-DPG还可以用来解决其他问题，如DPG方法的最优控制。
In the numerical solution of partial differential equations (PDEs), a central question is the one of building variational formulations that are inf-sup stable not only at the infinite-dimensional level, but also at the finite-dimensional one. This guarantees that residuals can be used to tightly bound errors from below and above and is crucial for a posteriori error control and the development of adaptive strategies. In this framework, the so-called Discontinuous Petrov--Galerkin (DPG) concept can be viewed as a systematic strategy of contriving variational formulations which possess these desirable stability properties, see e. g. Broersen et al. [2015]. In this paper, we present a C++ library, Dune-DPG, which serves to implement and solve such variational formulations. The library is built upon the multipurpose finite element package Dune (see Blatt et al. [2016]). One of the main features of Dune-DPG is its flexibility which is achieved by a highly modular structure. The library can solve in practice some important classes of PDEs (whose range goes beyond classical second order elliptic problems and includes e. g. transport dominated problems). As a result, Dune-DPG can also be used to address other problems like optimal control with the DPG approach.