$$L^p$$ error estimates for numerical approximation of convex Hamilton–Jacobi equations

Abstract

Abstract We establish $$L^p$$ L p error estimates for monotone numerical schemes approximating convex Hamilton–Jacobi equations on the d -dimensional torus. Using the adjoint method and semiconcavity estimates, we first prove an $$L^1$$ L 1 error bound of order one for classical monotone schemes of Crandall–Lions type and semi-Lagrangian schemes under standard convexity assumptions on the Hamiltonian and semiconcavity assumptions on the initial datum. By interpolation with the classical $$L^\infty $$ L ∞ estimate, we obtain $$L^p$$ L p estimates for every $$1\le p<+\infty $$ 1 ≤ p < + ∞ .