An Operator-Splitting Scheme for Viscosity Solutions of Constrained Second-Order PDEs

This work introduces a novel operator-splitting finite element scheme for approximating viscosity solutions of constrained second-order partial differential equations (PDEs). The method is designed for solutions with low regularity in \( C(\overline{\Omega}_T) \cap H^1(\Omega_T) \). By decoupling the PDE evolution from constraint enforcement, the scheme combines a stabilized finite element method with a semi-implicit time-stepping strategy. We prove that the scheme satisfies a discrete comparison principle by showing the discrete operator yields an M-matrix, which guarantees monotonicity and \(L^\infty\)-stability. These properties, along with consistency, ensure convergence to the unique viscosity solution via the Barles-Souganidis framework. For regular solutions, we establish an optimal-order error estimate of \( O(\Delta t + h^2) \). Numerical experiments validate the theory.