Analysis of a fully-discrete, non-conforming approximation of evolution equations and applications

TL;DR

This paper proves the convergence of a fully-discrete non-conforming approximation method for evolution equations, providing a unified framework applicable to various problems including the p-Navier-Stokes equations, with numerical validation.

Contribution

It introduces an abstract convergence framework for non-conforming Bochner pseudo-monotone operators, unifying analysis of multiple evolution problems and discretization schemes.

Findings

Convergence of discrete solutions to weak solutions is established.

Framework applies to various evolution problems, including p-Navier-Stokes.

Numerical experiments validate theoretical results.

Abstract

In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a 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. Therefore, the result can be interpreted either as a justification of the numerical method or as an alternative way of constructing weak solutions. We formulate the problem in the very general and abstract setting of so-called non-conforming Bochner pseudo-monotone operators, which allows for a unified treatment of several evolution problems. Our abstract results for non-conforming Bochner pseudo-monotone operators allow to establish (weak) convergence just by verifying a few natural assumptions on the operators time-by-time and on the discretization spaces. Hence, applications and extensions to several other evolution problems can be performed easily. We exemplify the applicability of our approach on several DG schemes for the unsteady $p$-Navier-Stokes problem. The results of some numerical experiments are reported in the final section.