The finite element approximation of Hamilton-Jacobi-Bellman equations

L∞-error estimates of finite element methods with Euler time discretization scheme for an evolutionary HJB equations with nonlinear source terms

“…The sequences ( ) k h Û well defined in K and converge to the unique solution of system of inequalities (8). where…”

“…At each iteration, a linear complementary subproblem or a linear equation system subproblem is solved. Boulbrachene and Haiour [8], by means of a subsolution method, conducted a finite element approximation study for the first time, for the stationary of the problem (1) and by using Bensoussan-Lions algorithm [4], a quasi-optimal error estimate in the ∞ L -norm has been derived according the following result…”

“…In this paper, an ∞ L -error estimate is established combining the geometric convergence of discrete iterative schemes using the known ∞ L -error estimates for stationary and evolutionary free boundary problems (cf., e.g., [2,8]) which play a major role in the finite element error analysis section. Finally the asymptotic behavior in uniform norm is deduced which investigated the evolutionary free boundary problem similar to that in [1].…”

“…In Section 2 and 3, we consider the discrete system of quasi-variational inequalities, discretize the iterative scheme by the standard finite element method combined with a theta scheme and an algorithm iterative discrete scheme is introduced. Then its geometric convergence is proved with respect to ∞ L -stability of the solution and the right-hand side and its characterization as the least upper bound of the subsolutions set (see also [1,8]). It is worth mentioning that this approach is entirely different from the one developed for the evolutionary problem.…”

A New Mathematical Model of Heat Equations And its Application on the Agriculture Soil

“…Finite element method is one of the ways of applied Mathematics. And we put it in space, using the principles inherited from the transformative formulation or weak wording, separate mathematical algorithm to find approximate solution to the problem of free boundary (see [3], [4] and [14]). It is said Dirichlet conditions or Neumann data.…”