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Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method
Advances in Computational Mathematics ( IF 1.7 ) Pub Date : 2020-12-17 , DOI: 10.1007/s10444-020-09827-6
Moreno Pintore , Federico Pichi , Martin Hess , Gianluigi Rozza , Claudio Canuto

The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.



中文翻译:

缩小基础谱元素法的紧缩方法有效计算分叉图

可以使用偏微分方程(PDE)来描述大多数最常见的物理现象。但是,它们通常具有强烈的非线性特征。这些特征导致分叉理论研究的多种解决方案并存。不幸的是,在实际情况下,即使存在更多分支时,即使经典技术通常仅能够为参数的任何值计算单个解,也必须利用数值方法来计算PDE系统的解。在这项工作中,我们实现了一种精心设计的紧缩放气连续法,该方法依靠光谱元素法(SEM)和基于简化基(RB)的方法来有效地计算具有更多参数和更多分叉点的分叉图。放气的连续方法可以结合经典的连续方法和放气方法:前者用于完全跟踪图中的每个已知分支,而后者则用于发现新的分支。最后,当考虑多个参数时,可以在在线阶段计算图表,而在离线阶段仅计算一维图表,从而确保了计算效率。在这项工作中,在对该方法进行更详细的描述之后,我们将显示使用该方法来计算与由Navier-Stokes方程控制的问题相关的分叉图所获得的结果。而利用后者来发现新的。最后,当考虑多个参数时,可以在在线阶段计算图表,而在离线阶段仅计算一维图表,从而确保了计算效率。在这项工作中,在对该方法进行更详细的描述之后,我们将显示使用该方法来计算与由Navier-Stokes方程控制的问题相关的分叉图所获得的结果。而利用后者来发现新的。最后,当考虑多个参数时,可以在在线阶段计算图表,而在离线阶段仅计算一维图表,从而确保了计算效率。在这项工作中,在对该方法进行更详细的描述之后,我们将显示使用该方法来计算与由Navier-Stokes方程控制的问题相关的分叉图所获得的结果。

更新日期:2020-12-21
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