常微分方程的多自适应Galerkin方法III：先验误差估计
Multiadaptive Galerkin Methods for ODEs III: A Priori Error Estimates
AI总结：
针对常微分方程初值问题，提出多自适应连续/间断Galerkin方法mcG(q)和mdG(q)，并证明其一般阶先验误差估计。
摘要：
用于常微分方程初值问题数值求解的多自适应连续/间断Galerkin方法mcG(q)和mdG(q)基于时间分区上的q次分段多项式逼近，其中时间步长可随计算解的不同分量而变化。本文证明了mcG(q)和mdG(q)方法的一般阶先验误差估计。为证明误差估计，我们将误差表示为离散对偶解与精确解插值余项的形式。然后通过插值估计以及离散对偶解的稳定性估计得到误差估计。
英文摘要：
The multiadaptive continuous/discontinuous Galerkin methods mcG(q) and mdG(q) for the numerical solution of initial value problems for ordinary differential equations are based on piecewise polynomial approximation of degree q on partitions in time with time steps which may vary for different components of the computed solution. In this paper, we prove general order a priori error estimates for the mcG(q) and mdG(q) methods. To prove the error estimates, we represent the error in terms of a discrete dual solution and the residual of an interpolant of the exact solution. The estimates then follow from interpolation estimates, together with stability estimates for the discrete dual solution.
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