A convergence theory for differentiable non-monotone schemes for fully nonlinear parabolic equations
This work addresses a theoretical gap for researchers in numerical analysis and PDEs by enabling convergence analysis for non-monotone differentiable schemes, though it is incremental as it builds on existing frameworks.
The paper tackles the convergence of differentiable approximation schemes for fully nonlinear parabolic PDEs, which are non-monotone and not covered by classical theory, by introducing an abstract framework with approximate monotonicity and weak stability conditions, leading to qualitative convergence and quantitative error estimates for specific equations.
This paper studies convergence of differentiable approximation schemes for terminal value problems of fully nonlinear parabolic partial differential equations. Differentiable schemes approximate spatial derivatives directly via gradients of smooth basis functions and are therefore generally non-monotone, placing them outside the scope of the classical Barles--Souganidis convergence theory. To address this, we introduce an abstract framework for differentiable schemes and establish convergence under conditions we call approximate monotonicity and weak stability. A central tool is a max-min representation of the nonlinearity, which allows us to relax the strict monotonicity condition when the approximate solution is smooth. We apply the abstract framework to kernel-based function approximation methods and derive qualitative convergence for general equations and quantitative error estimates for Hamilton--Jacobi--Bellman equations. Numerical experiments support the theoretical results and confirm the computational feasibility of the proposed approach.