Iain Smears

On the convergence of finite element methods for Hamilton-Jacobi-Bellman equations

We study the convergence of monotone

Discontinuous Galerkin Finite Element Approximation of Hamilton--Jacobi--Bellman Equations with Cordes Coefficients

P1

Discontinuous Galerkin Finite Element Approximation of Nondivergence Form Elliptic Equations with Cordès Coefficients

finite element methods on unstructured meshes for fully nonlinear Hamilton--Jacobi--Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretizations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under nondegeneracy assumptions, strong

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

L^2

Stable Discontinuous Galerkin FEM Without Penalty Parameters

convergence of the gradients.

Guaranteed, Locally Space-Time Efficient, and Polynomial-Degree Robust a Posteriori Error Estimates for High-Order Discretizations of Parabolic Problems

We propose an

Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method

hp

Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations

-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman equations with Cordes coefficients. The method is proved to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme. An existence and uniqueness result for strong solutions of the fully nonlinear problem and a semismoothness result for the nonlinear operator are also provided.

Nonoverlapping Domain Decomposition Preconditioners for Discontinuous Galerkin Approximations of Hamilton–Jacobi–Bellman Equations

Nondivergence form elliptic equations with discontinuous coefficients do not generally possess a weak formulation, thus presenting an obstacle to their numerical solution by classical finite element methods. We propose a new

Discrete p-robust \varvec{H}({{\mathrm{div}}})-liftings and a posteriori estimates for elliptic problems with H^{-1} source terms

hp