102 research outputs found
Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces
Based on a geometric discretization scheme for Maxwell equations, we unveil a
mathematical\textit{\}transformation between the electric field intensity
and the magnetic field intensity , denoted as Galerkin duality. Using
Galerkin duality and discrete Hodge operators, we construct two system
matrices, (primal formulation) and (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.Comment: 22 pages and 4 figure
Analysis of Compatible Discrete Operator Schemes for the Stokes Equations on Polyhedral Meshes
Compatible Discrete Operator schemes preserve basic properties of the
continuous model at the discrete level. They combine discrete differential
operators that discretize exactly topological laws and discrete Hodge operators
that approximate constitutive relations. We devise and analyze two families of
such schemes for the Stokes equations in curl formulation, with the pressure
degrees of freedom located at either mesh vertices or cells. The schemes ensure
local mass and momentum conservation. We prove discrete stability by
establishing novel discrete Poincar\'e inequalities. Using commutators related
to the consistency error, we derive error estimates with first-order
convergence rates for smooth solutions. We analyze two strategies for
discretizing the external load, so as to deliver tight error estimates when the
external load has a large irrotational or divergence-free part. Finally,
numerical results are presented on three-dimensional polyhedral meshes
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