Analysis of fully discrete, quasi non-conforming approximations of evolution equations and applications

TL;DR

This paper proves the convergence of fully discrete quasi non-conforming approximations for abstract evolution equations, providing a unified framework applicable to fluid dynamics and other problems, with numerical validation.

Contribution

It introduces a general convergence result for quasi non-conforming discretizations of evolution equations using pseudo-monotone operators, unifying various applications.

Findings

Convergence of discrete solutions to weak solutions is established.

The framework applies to incompressible fluid motion problems.

Numerical experiments support theoretical results.

Abstract

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.