论文标题
用于梯度流动的有限元方法的变异分析
A variational analysis for the moving finite element method for gradient flows
论文作者
论文摘要
通过使用Onsager原理作为近似工具，我们为梯度流动方程的移动有限元方法提供了一种新颖的推导。我们表明，离散的问题具有与连续的问题相同的能量耗散结构。这使我们能够使用自由结分段多项式的近似理论对非线性反应扩散方程的固定解进行数值分析。我们表明，在某些条件下，通过移动有限元方法获得的解决方案在时间到达无穷大时会收敛到总能量的局部最小化器。一旦通过离散方案检测到全局最小化器，就以最佳顺序近似连续的固定解决方案。给出了线性扩散方程和非线性allen-cahn方程的数值示例以验证分析结果。
By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.
