TITLE:
Helmholtz Decomposition of Vector Fields Using an Optimal Preconditioned Conjugate Gradient Algorithm
AUTHORS:
Jorge Lopez
KEYWORDS:
Helmholtz Decomposition, Self-Adjoint Operator, Optimal Preconditioning, Finite Element
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.11 No.5,
May
25,
2023
ABSTRACT: In this article, we study numerically a Helmholtz decomposition methodology, based on a formulation of the mathematical model as a saddle-point problem. We use a preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To solve the elliptic partial differential equations, we use a second order mixed finite element approximation for discretization. We show, using 2-D synthetic vector fields, that this approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation.