An optimal-order error estimate is presented for the arbitrary Lagrangian–Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerical solution of the Laplace equation with specified boundary value matching the velocity of the boundary. The optimal order of convergence is obtained by comparing the numerical solution with the ALE-Ritz projection of the exact solution, and by establishing an optimal-order estimate for the material derivative of the ALE-Ritz projection error.

中文翻译：

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演化域中抛物线方程的任意拉格朗日-欧拉等参数有限元方法的最优收敛

对演化域中抛物线方程的任意拉格朗日-欧拉 (ALE) 有限元方法提出了最优阶误差估计，使用域内部具有平面单纯形的高阶等参有限元。网格速度可以是给定体积速度场的线性近似，也可以是具有与边界速度相匹配的指定边界值的拉普拉斯方程的数值解。通过将数值解与精确解的 ALE-Ritz 投影进行比较，并通过建立 ALE-Ritz 投影误差的材料导数的最优阶估计来获得收敛的最优阶。