A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems
A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems
复制标题
波浪问题时空Trefftz间断伽辽金方法的先验误差分析
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
S. Schnepp
中科院分区：
文献类型：
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作者：
F. Kretzschmar;A. Moiola;I. Perugia;S. Schnepp
提出并分析了波传播问题的时空间断Galerkin方法。该计划的特点是，它是一个Trefftz方法，即试验和测试功能的解决方案的偏微分方程离散在每个元素的（时空）网格。所考虑的方法是对Kretzschmar等人（2014）和Monk & Richter（2005）的间断Galerkin格式的修改。对于一维空间中的麦克斯韦方程，我们证明了该方法的稳定性、拟最优性、多项式Trefftz空间的最佳逼近估计以及在网格宽度和多项式次数上具有高阶的（全显式）误差界.
分析框架也适用于标量波问题和麦克斯韦方程组在更高的空间维度。一些数值实验证明了理论结果证明和更快的收敛速度相比，non-Trefftz版本的计划。
We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree.
The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.