38 research outputs found
Discontinuous Galerkin finite element methods for time-dependent Hamilton--Jacobi--Bellman equations with Cordes coefficients
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 - and -refinement.Comment: 40 pages, 3 figures, submitted; extended version with supporting
appendi
Adaptive C\u3csup\u3e0\u3c/sup\u3e interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients
In this paper we conduct a priori and a posteriori error analysis of the C interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H . 0
Adaptive interior penalty methods for Hamilton–Jacobi–Bellman equations with Cordes coefficients
In this paper we conduct a priori and a posteriori error analysis of the C0 interior penalty method for Hamilton–Jacobi–Bellman equations, with coefficients that satisfy the Cordes condition. These estimates show the quasi-optimality of the method, and provide one with an adaptive finite element method. In accordance with the proven regularity theory, we only assume that the solution of the Hamilton–Jacobi–Bellman equation belongs to H2
Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations
We analyse a class of nonoverlapping domain decomposition preconditioners for
nonsymmetric linear systems arising from discontinuous Galerkin finite element
approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial
differential equations. These nonsymmetric linear systems are uniformly bounded
and coercive with respect to a related symmetric bilinear form, that is
associated to a matrix . In this work, we construct a
nonoverlapping domain decomposition preconditioner , that is based
on , and we then show that the effectiveness of the preconditioner
for solving the} nonsymmetric problems can be studied in terms of the condition
number . In particular, we establish the
bound , where
and are respectively the coarse and fine mesh sizes, and and
are respectively the coarse and fine mesh polynomial degrees. This represents
the first such result for this class of methods that explicitly accounts for
the dependence of the condition number on ; our analysis is founded upon an
original optimal order approximation result between fine and coarse
discontinuous finite element spaces. Numerical experiments demonstrate the
sharpness of this bound. Although the preconditioners are not robust with
respect to the polynomial degree, our bounds quantify the effect of the coarse
and fine space polynomial degrees. Furthermore, we show computationally that
these methods are effective in practical applications to nonsymmetric, fully
nonlinear HJB equations under -refinement for moderate polynomial degrees
Mixed finite element approximation of periodic Hamilton--Jacobi--Bellman problems with application to numerical homogenization
In the first part of the paper, we propose and rigorously analyze a mixed
finite element method for the approximation of the periodic strong solution to
the fully nonlinear second-order Hamilton--Jacobi--Bellman equation with
coefficients satisfying the Cordes condition. These problems arise as the
corrector problems in the homogenization of Hamilton--Jacobi--Bellman
equations. The second part of the paper focuses on the numerical homogenization
of such equations, more precisely on the numerical approximation of the
effective Hamiltonian. Numerical experiments demonstrate the approximation
scheme for the effective Hamiltonian and the numerical solution of the
homogenized problem.Comment: 23 page