A Petrov-Galerkin Finite Element Scheme for 1-D Time-independent Hamilton-Jacobi-Bellman Equations

Abstract

A numerical method for solving 1-D time-independent Hamilton-Jacobi-Bellman equations, which are referred to as 1-D HJBEs, is presented and applied to test cases for assessing its computational performance. An HJBE in this paper is a nonconservative second-order ordinary differential equation having linear diffusion and nonlinear drift terms. This paper applies a regularization method to the drift coefficient of the HJBEs, which helps well-pose the boundary value problems of the equations in the classical sense. A mathematical analysis on consistency errors between the solutions to the original and regularized HJBEs is performed. The derived results of the analysis show that the regularization method is mathematically consistent. The regularized HJBEs are solved with a Petrov-Galerkin finite element scheme, which is referred to as the PGFE scheme. The scheme is based on the fitting technique and is unconditionally stable for linear problems. Application of the scheme with the regularization method to the HJBEs with bounded drift coefficients demonstrates its satisfactory high computational accuracy. The optimal regularization parameter value as a function of the element size is then numerically identified. The computational results show that the PGFE scheme without the regularization method would fail to accurately capture solution profiles even if thousands of elements are used, which is not the case for the scheme with the regularization method even with hundreds of elements. Impacts of incorporating an adaptive re-meshing method, which is the moving mesh partial differential equation method, into the PGFE scheme are also assessed, demonstrating that it can enhance robustness of the scheme with regularization.

Optimization model to start harvesting in stochastic aquaculture system

“…The reduced BE is numerically solved to find the free boundary C that separates the spatio‐temporal domain Q into a waiting region S 1 and a harvesting region S 2 . The Petrov–Galerkin FEM (PGFEM) for numerical resolution of 1‐D parabolic PDEs , which has already been successfully applied to HJBEs arising in environmental and ecological problems , is employed in this paper. Utilizing the re‐interpretation of the reduced BE pointed out in Remark , the PGFEM can efficiently detect C .…”