Residual-Based Variational Multiscale Modeling in a Discontinuous Galerkin Framework
Residual-Based Variational Multiscale Modeling in a Discontinuous Galerkin Framework
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DOI：
10.1137/17m1147044
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发表时间：
2017-09
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影响因子：
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通讯作者：
S. Stoter;S. Turteltaub;S. Hulshoff;D. Schillinger
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文献类型：
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作者：
S. Stoter;S. Turteltaub;S. Hulshoff;D. Schillinger
我们在间断Galerkin框架下发展了变分多尺度方法的一般形式。我们的方法是基于真正的解决方案分解成不连续的粗尺度和不连续的细尺度部分。所得到的粗尺度弱公式包括两种类型的细尺度贡献。第一种类型对应于一个细尺度的体积项，我们制定了一个基于残差的模型，也考虑到在元素界面的细尺度效应。第二种类型由独立的细尺度条款在元素接口，我们制定了一个新的细尺度“接口模型”。我们证明了一维泊松问题，现有的不连续Galerkin公式，如内部惩罚方法，可以通过选择特定的细尺度界面模型重新推导。因此，多尺度制定打开了一个新的视角不连续Galerkin方法及其数值特性的大门。这是证明了一维对流扩散问题，在那里我们表明，逆风数值通量可以被解释为一个特设的补救措施缺失体积细尺度条款。
We develop the general form of the variational multiscale method in a discontinuous Galerkin framework. Our method is based on the decomposition of the true solution into discontinuous coarse-scale and discontinuous fine-scale parts. The obtained coarse-scale weak formulation includes two types of fine-scale contributions. The first type corresponds to a fine-scale volumetric term, which we formulate in terms of a residual-based model that also takes into account fine-scale effects at element interfaces. The second type consists of independent fine-scale terms at element interfaces, which we formulate in terms of a new fine-scale "interface model". We demonstrate for the one-dimensional Poisson problem that existing discontinuous Galerkin formulations, such as the interior penalty method, can be rederived by choosing particular fine-scale interface models. The multiscale formulation thus opens the door for a new perspective on discontinuous Galerkin methods and their numerical properties. This is demonstrated for the one-dimensional advection-diffusion problem, where we show that upwind numerical fluxes can be interpreted as an ad hoc remedy for missing volumetric fine-scale terms.