Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations
Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations
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DOI：
10.1016/j.cam.2015.10.024
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发表时间：
2016-06-01
影响因子：
2.4
通讯作者：
Neilan, Michael
中科院分区：
文献类型：
--
作者：
Feng, Xiaobing;Lewis, Thomas;Neilan, Michael
建立了逼近Sobolev函数和分段Sobolev函数弱导数的间断Galerkin（DG）有限元微分理论。通过引入数值单边导数作为构造块，定义了梯度、散度、Hessian算子和Laplacian算子等一阶和二阶数值算子，并建立了相应的微积分规则.微积分的规则包括乘积和链式规则、分部积分公式和发散定理。在笛卡尔网格上建立了DG有限元数值导数与一些著名的有限差分数值导数公式之间的关系和逼近性质。除了对数值微分的兴趣之外，发展DG有限元微分学的主要动机和目标是求解偏微分方程。结果表明，几个现有的有限元，有限差分和间断伽辽金方法可以重写compressible使用建议DG有限元微分框架。此外，在此框架下，还得到了求解线性和非线性偏微分方程的新的间断Galerkin方法。(C)2015 Elsevier B.V.版权所有。
This paper develops a discontinuous Galerkin (DG) finite element differential calculus theory for approximating weak derivatives of Sobolev functions and piecewise Sobolev functions. By introducing numerical one-sided derivatives as building blocks, various first and second order numerical operators such as the gradient, divergence, Hessian, and Laplacian operator are defined, and their corresponding calculus rules are established. Among the calculus rules are product and chain rules, integration by parts formulas and the divergence theorem. Approximation properties and the relationship between the proposed DG finite element numerical derivatives and some well-known finite difference numerical derivative formulas on Cartesian grids are also established. Besides independent interest in numerical differentiation, the primary motivation and goal of developing the DG finite element differential calculus is to solve partial differential equations. It is shown that several existing finite element, finite difference and discontinuous Galerkin methods can be rewritten compactly using the proposed DG finite element differential calculus framework. Moreover, new discontinuous Galerkin methods for linear and nonlinear PDEs are also obtained from the framework. (C) 2015 Elsevier B.V. All rights reserved.