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

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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 for both rough and regular solutions in terms of temporal regularity show that 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 achieved under combined hp- and \(\tau q\)-refinement.

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Smears, I., Süli, E. Discontinuous Galerkin finite element methods for time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients. Numer. Math. 133, 141–176 (2016). https://doi.org/10.1007/s00211-015-0741-6

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  • DOI: https://doi.org/10.1007/s00211-015-0741-6

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