发展方程的全离散拟非协调逼近分析及应用
Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications
摘要：
本文考虑了抽象发展方程的全离散逼近，采用拟非协调空间近似和时间有限差分（Rothe-Galerkin方法）。主要结果是离散解收敛到连续问题的一个弱解。因此，该结果既可以解释为数值方法的合理性，也可以解释为构造弱解的另一种方法。我们把问题的伪单调算子，它允许一个统一的治疗几个演化问题的非常一般和抽象的设置。的例子-适合我们的设置，这激励了我们的研究-是描述不可压缩流体的运动的问题，因为准非协调近似允许处理问题规定的分歧。我们的抽象结果的伪单调算子允许显示收敛只需验证一些自然的假设上的操作时间和离散空间。因此，可以很容易地执行其他几个进化问题的应用程序和扩展。一些数值实验的结果报告在最后一节。
英文摘要：
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.
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