# Discontinuous Galerkin finite element methods for time-dependent Hamilton-Jacobi-Bellman equations with Cordes coefficients

Abstract : We propose and analyse a fully-discrete discontinuous Galerkin time-stepping method for parabolic Hamilton–Jacobi–Bellman equations with Cordes coefficients. The method is consistent and unconditionally stable on rather general unstructured meshes and time-partitions. Error bounds are obtained for both rough and regular solutions, and it is shown that for sufficiently smooth solutions, the method is arbitrarily high-order with optimal convergence rates with respect to the mesh size, time-interval length and temporal polynomial degree, and possibly suboptimal by an order and a half in the spatial polynomial degree. Numerical experiments on problems with strongly anisotropic diffusion coefficients and early-time singularities demonstrate the accuracy and computational efficiency of the method, with exponential convergence rates under combined $h p$-and $\tau q$-refinement.
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https://hal.inria.fr/hal-01428647
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Iain Smears, Endre Süli. Discontinuous Galerkin finite element methods for time-dependent Hamilton-Jacobi-Bellman equations with Cordes coefficients. Numerische Mathematik, Springer Verlag, 2016, 133 (1), pp.141 - 176. ⟨10.1007/s00211-015-0741-6⟩. ⟨hal-01428647⟩

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