Finite element approximation of invariant manifolds by the parameterization method
Finite element approximation of invariant manifolds by the parameterization method
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DOI：
10.1007/s42985-022-00214-y
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发表时间：
2022-03
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通讯作者：
Jorge Gonzalez;J. D. M. James;N. Tuncer
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作者：
Jorge Gonzalez;J. D. M. James;N. Tuncer
将不变流形的参数化方法与椭圆型偏微分方程的有限元方法相结合，得到了非线性抛物型偏微分方程不稳定平衡解的不变流形高阶近似的一个新的计算框架.参数化方法提供了一个无穷小不变方程的不变流形，我们解决了通过一个幂级数分析。一个幂匹配的论点导致一个递归系统的线性椭圆型偏微分方程-所谓的同调方程-其解决方案是幂级数系数的参数化。同调方程递归求解到任何所需的阶数N使用有限元逼近。最终的结果是一个N阶多项式近似的图表映射的流形，在一个适当的有限元空间的系数。我们实现了各种示例问题的方法具有多项式和非多项式的非线性，非凸的二维多边形域（没有必要简单连接），平衡的解决方案与莫尔斯指数1和2。我们实现了后验误差指标，提供了数值证据，支持索赔的流形计算准确。
We combine the parameterization method for invariant manifolds with the finite element method for elliptic PDEs, to obtain a new computational framework for high order approximation of invariant manifolds attached to unstable equilibrium solutions of nonlinear parabolic PDEs. The parameterization method provides an infinitesimal invariance equation for the invariant manifold, which we solve via a power series ansatz. A power matching argument leads to a recursive systems of linear elliptic PDEs—the so called homological equations—whose solutions are the power series coefficients of the parameterization. The homological equations are solved recursively to any desired orderNusing finite element approximation. The end result is anN-th order polynomial approximation of a chart map of the manifold, with coefficients in an appropriate finite element space. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two dimensional polygonal domains (not necessary simply connected), for equilibrium solutions with Morse indices one and two. We implement a-posteriori error indicators which provide numerical evidence in support of the claim that the manifolds are computed accurately.