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 $hp$- and $\tau q$-refinement.

40 pages, 3 figures, submitted; extended version with supporting appendix

References in corpus (3)

• A finite element method for fully nonlinear elliptic problems
• On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations
• A finite element method for second order nonvariational elliptic problems

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