$\mathcal{K}-$convergence as a new tool in numerical analysis
Provides a new theoretical tool for proving convergence of numerical methods for nonlinear PDEs with minimal regularity, but the application is domain-specific to fluid dynamics.
The paper adapts K-convergence of Young measures to analyze numerical schemes, proving pointwise convergence of approximate solutions under minimal regularity, and applies it to a finite volume method for the isentropic Euler system, presenting it as a nonlinear Lax equivalence theorem.
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.