发表机构
Baylor University（贝勒大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对时变偏微分方程的有限元方法无法满足边界约束的问题，提出两种满足边界约束的高阶方法，并通过数值算例验证了方法的有效性。
AI 中文摘要
有限元通过将变分问题限制在有限维近似空间中，为偏微分方程的数值求解提供了精确且高效的方法，但通常无法保证满足原问题固有的边界约束。本文针对时变问题提出两种满足边界约束的方法：第一种是通用投影方法，通过系统性修改任意抽象时间步长格式得到；第二种是整体式技术，以隐式单级Runge-Kutta方法和通用隐式多步方法的修正格式为原型。通过求解约束优化问题，可确保离散时间层的近似解满足边界约束，在空间和时间上均达到（形式上的）高阶精度。文中给出了线性热传导方程、平流方程及非线性Allen-Cahn方程的数值算例。
英文摘要
Finite elements provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not, in general, guarantee enforcement of bounds constraints inherent in the original problem. We propose two approaches to enforcing bounds constraints for time-dependent problems. First, we propose general projective methods which result from a systematic modification of any abstract time-stepping scheme. Second, we present a monolithic technique for which we take a modified formulation of implicit single-stage Runge-Kutta methods and general implicit multistep methods as prototypical examples. By solving a constrained optimization problem, we are able to ensure that the bounds constraints are enforced by the approximate solution at the discrete time levels, obtaining (formally) high order methods both in space and time. Numerical examples for the linear heat and advection equations and nonlinear Allen-Cahn equation are given.
Comments24 pages
