论文标题
Korteweg-de Vries方程的无条件稳定时空FE方法
An Unconditionally Stable Space-Time FE Method for the Korteweg-de Vries Equation
论文作者
论文摘要
我们引入了一种无条件稳定的有限元（Fe）方法，即用于Korteweg-De Vries（KDV）方程的数值分析的自动变异稳定Fe（AVS-FE）方法。 AVS-FE方法是一种Petrov-Galerkin方法，它采用了不连续的Petrov-Galerkin（DPG）方法的最佳不连续测试函数的概念。但是，由于AVS-FE方法是一种最小残留方法，因此我们建立了一个全局的鞍点系统，而不是计算最佳测试功能逐元元素。该系统允许我们同时寻求KDV初始边界值问题（IBVP）的近似解决方案和近似误差的Riesz代表。 AVS-FE方法通过使用全球连续的希尔伯特空间（例如H1）将自己与其他最小残留方法区分开来，同时使用破碎的希尔伯特空间进行测试。因此，AVS-FE近似是经典的C0连续Fe溶液。这种方法的无条件稳定性使我们能够解决KDV方程空间和时间，而不必满足CFL条件。我们为KDV方程的线性和非线性版本提供了几种数值验证，从而导致最佳收敛行为。最后，我们对非线性KDV方程的空间和时间上的自适应网格细化进行了数值验证。
We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg-de Vries (KdV) equation. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov- Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial boundary value problem (IBVP) and a Riesz representer of the approximation error. The AVS-FE method distinguishes itself from other minimum residual methods by using globally continuous Hilbert spaces, such as H1, while at the same time using broken Hilbert spaces for the test. Consequently, the AVS-FE approximations are classical C0 continuous FE solutions. The unconditional stability of this method allows us to solve the KdV equation space and time without having to satisfy a CFL condition. We present several numerical verifications for both linear and nonlinear versions of the KdV equation leading to optimal convergence behavior. Finally, we present a numerical verification of adaptive mesh refinements in both space and time for the nonlinear KdV equation.
