$\mathcal{K}-$convergence as a new tool in numerical analysis

TL;DR

This paper introduces $\\mathcal{K}$-convergence as a novel analytical tool to study the pointwise convergence of numerical solutions, especially for problems with minimal regularity, demonstrated through a finite volume method for fluid dynamics.

Contribution

It adapts $\\mathcal{K}$-convergence of Young measures to numerical schemes and establishes new convergence results for solutions with minimal regularity.

Findings

Proves pointwise convergence of numerical solutions under minimal regularity.

Applies the theory to a finite volume scheme for the Euler system.

Provides a nonlinear analogue of the Lax equivalence theorem.

Abstract

We adapt the concept of $\mathcal{K}-$convergence of Young measures to the sequences of approximate solutions resulting from numerical schemes. We obtain new results on pointwise convergence of numerical solutions in the case when solutions of the limit continuous problem possess minimal regularity. We apply the abstract theory to a finite volume method for the isentropic Euler system describing the motion of a compressible inviscid fluid. The result can be seen as a nonlinear version of the fundamental Lax equivalence theorem.