Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces
نویسنده
چکیده
Based on a geometric discretization scheme for Maxwell equations, we unveil a mathematical transformation between the electric field intensity E and the magnetic field intensity H, denoted as Galerkin duality. Using Galerkin duality and discrete Hodge operators, we construct two system matrices, [XE ] (primal formulation) and [XH ] (dual formulation) respectively, that discretize the second-order vector wave equations. We show that the primal formulation recovers the conventional (edge-element) finite element method (FEM) and suggests a geometric foundation for it. On the other hand, the dual formulation suggests a new (dual) type of FEM. Although both formulations give identical dynamical physical solutions, the dimensions of the null spaces are different. PACS numbers: 02.70.Dh; 03.50.De; 02.60.-x; 41.20.-q.
منابع مشابه
Superconvergent biquadratic finite volume element method for two-dimensional Poisson's equations
The authors consider the biquadratic finite volume element approximation for the Pois-son's equation on the rectangular domain Ω = (0, 1) 2. The primal mesh is performed using a ractangular partition. The control volumes are chosen in such a way that the vertices are stress points of the primal mesh. In order to solve the scheme more efficiently, the authors wrote the bi-quadratic finite volume...
متن کاملVARIATIONAL DISCRETIZATION AND MIXED METHODS FOR SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS WITH INTEGRAL CONSTRAINT
The aim of this work is to investigate the variational discretization and mixed finite element methods for optimal control problem governed by semi linear parabolic equations with integral constraint. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control is not discreted. Optimal error estimates in L2 are established for the state...
متن کاملFinite element exterior calculus, homological techniques, and applications
Finite element exterior calculus is an approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretizations which are compatible with the geometric, topological, and algebraic structures which...
متن کاملGeometric Aspects of the Simplicial Discretization of Maxwell’s Equations
Aspects of the geometric discretization of electromagnetic fields on simplicial lattices are considered. First, the convenience of the use of exterior differential forms to represent the field quantities through their natural role as duals (cochains) of the geometric constituents of the lattice (chains = nodes, edges, faces, volumes) is briefly reviewed. Then, the use of the barycentric subdivi...
متن کاملBlock Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD
The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is very challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible ...
متن کامل