Finite volume approximation of the Maxwell's equations in nonhomogeneous media.

Chung Tsz Shun Eric.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 102-104).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Applications of Maxwell's equations --- p.1Chapter 1.2 --- Introduction to Maxwell's equations --- p.2Chapter 1.3 --- Historical outline of numerical methods --- p.4Chapter 1.4 --- A new approach --- p.5Chapter 2 --- Mathematical Backgrounds --- p.7Chapter 2.1 --- Sobolev spaces --- p.7Chapter 2.2 --- Tools from functional analysis --- p.8Chapter 3 --- Discretization of Vector Fields --- p.10Chapter 3.1 --- Domain triangulation --- p.10Chapter 3.2 --- Mesh dependent norms --- p.11Chapter 3.3 --- Discrete circulation operators --- p.13Chapter 3.4 --- Discrete flux operators --- p.20Chapter 4 --- Spatial Discretization of the Maxwell's Equations --- p.23Chapter 4.1 --- Derivation --- p.23Chapter 4.2 --- Consistency theory --- p.29Chapter 4.3 --- Convergence theory --- p.33Chapter 4.3.1 --- Polyhedral domain --- p.33Chapter 4.3.2 --- Rectangular domain --- p.38Chapter 5 --- Fully Discretization of the Maxwell's Equations --- p.63Chapter 5.1 --- Derivation --- p.63Chapter 5.2 --- Consistency theory --- p.65Chapter 5.3 --- Convergence theory --- p.69Chapter 5.3.1 --- Polyhedral domain --- p.69Chapter 5.3.2 --- Rectangular domain --- p.77Chapter 6 --- Numerical Tests --- p.97Chapter 6.1 --- Convergence test --- p.97Chapter 6.2 --- Electromagnetic scattering --- p.99Bibliography --- p.10

Staggered discontinuous Galerkin method for the curl-curl operator and convection-diffusion equation.

Lee, Chak Shing."August 2011."Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (leaves 60-62).Abstracts in English and Chinese.Chapter 1 --- Model Problems --- p.1Chapter 1.1 --- Introduction --- p.1Chapter 1.2 --- The curl-curl operator --- p.2Chapter 1.3 --- The convection-diffusion equation --- p.6Chapter 2 --- Staggered DG method for the Curl-Curl operator --- p.8Chapter 2.1 --- Introduction --- p.8Chapter 2.2 --- Discontinuous Galerkin discretization --- p.8Chapter 2.3 --- Stability for aligned fields --- p.14Chapter 2.4 --- Error estimates --- p.17Chapter 2.5 --- Numerical experiments --- p.21Chapter 2.6 --- Concluding Remarks --- p.32Chapter 3 --- Staggered DG method for the convection-diffusion equation --- p.33Chapter 3.1 --- Introduction --- p.33Chapter 3.2 --- Method description --- p.33Chapter 3.3 --- Preservation of physical structures --- p.38Chapter 3.4 --- Stability and convergence --- p.42Chapter 3.4.1 --- Static problem --- p.42Chapter 3.4.2 --- Time-dependent problem --- p.46Chapter 3.5 --- Fully discrete scheme --- p.49Chapter 3.6 --- Numerical examples --- p.55Chapter 3.6.1 --- The static problem --- p.55Chapter 3.6.2 --- Time dependent problem --- p.56Chapter 3.7 --- Concluding Remark --- p.59Bibliography --- p.6

The convergence of mimetic discretization for rough grids

AbstractWe prove that the mimetic finite-difference discretizations of Laplace's equation converges on rough logically-rectangular grids with convex cells. Mimetic discretizations for the invariant operators' divergence, gradient, and curl satisfy exact discrete analogs of many of the important theorems of vector calculus. The mimetic discretization of the Laplacian is given by the composition of the discrete divergence and gradient. We first construct a mimetic discretization on a single cell by geometrically constructing inner products for discrete scalar and vector fields, then constructing a finite-volume discrete divergence, and then constructing a discrete gradient that is consistent with the discrete divergence theorem. This construction is then extended to the global grid. We demonstrate the convergence for the two-dimensional Laplace equation with Dirichlet boundary conditions on grids with a lower bound on the angles in the cell corners and an upper bound on the cell aspect ratios. The best convergence rate to be expected is first order, which is what we prove. The techniques developed apply to far more general initial boundary-value problems

Etude d'une méthode de volumes finis pour la résolution des équations de Maxwell en deux dimensions d'espace sur des maillages quelconques et couplage avec l'équation de Vlasov

Nous développons et étudions une méthode de volumes finis pour résoudre le système de Maxwell instationnaire bidimensionnel sur des maillages presque quelconques (non-conformes, non-convexes, aplatis..). Nous commençons par la construction du schéma, qui est basé sur l'utilisation des opérateurs discrets de la méthode DDFV et sur un choix pertinent pour la discrétisation des conditions initiales et des conditions aux limites. Ensuite, nous prouvons que ce schéma préserve localement la condition de divergence, que l'énergie électromagnétique discrète est conservée ou décroissante (selon les conditions aux limites) et qu'elle est positive sous condition CFL. Nous montrons aussi la stabilité du schéma sous condition CFL et sa convergence dans les cas de champs réguliers et non réguliers. Ces résultats sont ensuite validés, numériquement avec quelques cas tests sur différents types de maillages. Nous vérifions aussi que l'utilisation des maillages non conformes n'amplifie pas les réflexions parasites. Enfin nous couplons ce schéma avec une méthode PIC pour résoudre le système de Maxwell-Vlasov. Nous calculons la densité de courant avec une généralisation de la méthode de Buneman à des maillages quelconques et nous montrons la conservation des équations de charge discrètes, ce qui permet de conserver la loi de Gauss. Le problème couplé est validé numériquement et la simulation de l'amortissement Landau confirme la décroissance de l'énergie, portée par le champ électrique, avec une précision dépendant du nombre de particules par mailleWe develop and study a finite volume method to solve the bidimensional nonstationary Maxwell equations on arbitrary (non-conforming, non-convex, flat...) meshes. We start by the construction of the scheme, which is based on the use of the DDFV discrete operators and a pertinent choice to discretize initial and boundary conditions. Then, we prove that the scheme locally preserves the divergence condition, that a discrete electromagnetic energy is conserved or decreasing (depending on boundary conditions) and that it is positive under a CFL condition. We also show the stability of the scheme under a CFL condition and its convergence for regular and non-regular fields. Then, these results are numerically validated with some tests using different types of meshes. We verify, also, that the use of non-conforming meshes doesn't amplify parasitic reflections. Finally, we coupled the scheme with a PIC method to solve the Maxwell-Vlasov system. We calculate the current density using a generalization of Buneman's method to arbitrary meshes and we prove that discrete charge equations, and thus Gauss' law, are conserved. The coupled problem is numerically validated and the simulation of Landau damping confirms the electric energy decrease with a precision depending on the number of particles per cel