论文标题
分析完全离散的，未符合进化方程和应用的不合格近似值
Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications
论文作者
论文摘要
在本文中，我们通过准符合的空间近似和时间上有限的差异（Rothe-Galerkin方法）考虑了抽象进化方程的完全离散近似值。主要结果是离散解决方案与连续问题的弱解决方案的收敛性。因此，结果可以解释为数值方法的理由，也可以将其解释为构建弱解决方案的替代方法。我们在伪超声酮操作员的非常通用和抽象的设置中设定了问题，这允许对几种进化问题进行统一的处理。这些示例 - 适合我们的设置并激发了我们的研究 - 是描述不可压缩流体运动的问题，因为准不合格近似可以处理规定的差异问题。我们对伪单酮操作员的抽象结果只能通过逐次验证操作员的一些自然假设和离散空间来显示收敛性。因此，可以轻松执行对其他几个进化问题的应用和扩展。最后一部分报告了一些数值内部的结果。
In this paper we consider fully discrete approximations of abstract evolution equations, by means of a quasi non-conforming spatial approximation and finite differences in time (Rothe-Galerkin method). The main result is the convergence of the discrete solutions to a weak solution of the continuous problem. Hence, the result can be interpreted either as a justification of the numerical method, or as an alternative way of constructing weak solutions. We set the problem in the very general and abstract setting of pseudo-monotone operators, which allows for a unified treatment of several evolution problems. The examples -- which fit into our setting and which motivated our research -- are problems describing the motion of incompressible fluids, since the quasi non-conforming approximation allows to handle problems with prescribed divergence. Our abstract results for pseudo-monotone operators allow to show convergence just by verifying a few natural assumptions on the operator time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be easily performed. The results of some numerical periments are reported in the final section.
