Error Bounds for Boundary-Stopped Positive Cubature Schemes for Hamilton--Jacobi--Bellman Equations

RESUMO

We derive error bounds for a boundary-stopped positive cubature scheme for degenerate parabolic Hamilton--Jacobi--Bellman equations with Dirichlet data on bounded domains. The scheme balances antipodal branches at their first boundary contacts and uses a control-dependent effective time. Its stopped-barrier remainder is \(O(\tau\Delta t^{\gamma/2})\), and the total mass of interpolated branches is at most \(2\tau/\Delta t\), where \(\tau\) is the effective time. After division by the effective time, these estimates give a consistency bound that is uniform as \(\tau\to0\). Under a common strict boundary barrier and the stated mesh conditions, interior consistency, coefficient shaking, switching, and localized comparison yield \[ \|(u-u_h)^+\|_\infty\le C(\Delta t^{1/4}+h\Delta t^{-1/2}), \qquad \|(u_h-u)^+\|_\infty\le C(\Delta t^{1/10}+h^{1/2}\Delta t^{-1/4}). \] Thus \(\Delta t\asymp h^{10/7}\) gives an \(O(h^{1/7})\) maximum-norm bound. The analysis applies to any fixed centrally symmetric positive degree-two cubature and permits rank-deficient diffusion. Numerical examples examine parallel heat loss to a cold wall and control-dependent, rank-deficient diffusion.